Philosophical Hinduism in its earliest form teaches a non-dualistic view of the universe, i.e., that there is no distinction between one's own Self (Atman), and the Ultimate Reality (Brahman, God).
The following are a few quotations from the Upanishads which teach this doctrine:
Mandukya Upanishad - Chapter 1
1 OM! This Imperishable Word is the whole of this visible universe. Its explanation is as follows: What has become, what is becoming, what will become - verily, all of this is OM. And what is beyond these three states of the world of time - that too, verily, is OM.
2 All this, verily, is Brahman. The Self is Brahman. This Self has four quarters.
Brihadaranyaka Upanishad - Book 1, Chapter 4
10 This (Self) was indeed Brahman in the beginning. It knew only Itself as, 'I am Brahman'. Therefore It became all. And whoever among the gods knew It also became That; and the same with sages and men. The sage Vamadeva, while realizing this (self) as That, knew, 'I was Manu, and the sun'. And to this day whoever in like manner knows It as, 'I am Brahman', becomes all this entire universe. Even the gods cannot prevail against him, for he becomes their self. While he who worships another god thinking, 'He is one, and I am another', does not know. He is like an animal to the gods.
Additionally, in the Upanishads, it is taught that enlightenment, and escape from the cycle of reincarnation (Samsara), is obtained by meditating on the fact that the Self is Brahman.
For instance:
Aitareya Upanishad - Chapter 3
1 Who is He upon whom we meditate thinking: "This is the Self"? Which one is the Self? Is it He by whom one sees form, by whom one hears sound and by whom one tastes the sweet and the unsweet?
2 Is it the heart (intellect) and the mind? It is consciousness, lordship, knowledge, wisdom, retentive power of mind, sense knowledge, steadfastness, thought, thoughtfulness, sorrow, memory, concepts, purpose, life, desire, longing: all these are but various names of Consciousness (Prajnanam).
3 He is Brahma, He is Indra, He is Prajapati; He is all these gods; He is the five great elements - earth, air, akasa, water, light; He is all these small creatures and the others which are mixed; He is the origin - those born of an egg, of a womb, of sweat and of a sprout; He is horses, cows, human beings, elephants - whatever breathes here, whether moving on legs or flying in the air or unmoving. All this is guided by Consciousness, is supported by Consciousness. The basis of the universe is Consciousness. Consciousness is Brahman (Prajnanam Brahma).
4 He, having realized oneness with Pure Consciousness, soared from this world and having obtained all desires in yonder heavenly world, became immortal - yea, became immortal.
Katha Upanishad - Book 2, Chapter 2
12 There is one Supreme Ruler, the inmost Self of all beings, who makes His one form manifold. Eternal happiness belongs to the wise, who perceive Him within themselves - not to others.
13 There is One who is the eternal Reality among non-eternal objects, the one truly conscious Entity among conscious objects and who, though non-dual, fulfils the desires of many. Eternal peace belongs to the wise, who perceive Him within themselves - not to others.
Katha Upanishad - Book 2, Chapter 3
18 Having received this wisdom taught by the King of Death and the entire process of yoga, Nachiketa became free from impurities and death and attained Brahman. Thus it will be also with any other who knows, in this manner, the inmost Self.
The ignorance of the truth that Atman (the Self) is Brahman is called "Maya", or illusion. In classical Hinduism, this is what keeps one locked in the cycle of rebirth.
However, this entire system is illogical, as if everything is the Ultimate Reality, there would be no reason for Maya to exist at all. Under such a system, the Ultimate Reality would have had to create Maya, and then embody itself in billions of manifestations, locking itself into a cycle of reincarnation in that form, with the goal of ultimately realizing that it is itself once again, and attaining the unity which it began with, which makes no sense whatsoever. It would be a concept of God which has God tormenting itself for no discernible reason.
Additionally, proposing that the Ultimate Reality is Atman would mean that there is no distinction whatsoever between Atman and Brahman. But identity (e.g. Atman=Brahman) requires that all essential properties be shared. Therefore, for anything to be Atman/Brahman, it would need to possess all of the essential properties of the Ultimate Reality - for instance, omniscience. Omniscience is not something "optional" for Atman/Brahman, but instead, an essential property. So, if something lacks omniscience - if it does not know all that can be known - then that thing is not Atman/Brahman. Therefore, since human beings and other conscious beings lack the properties essential to the Ultimate Reality, they are not the Ultimate Reality. And, if one were to say that these traits are locked behind Maya, or some other restriction, it is of no consequence, because the definition of omnipotence is not "could under some circumstances be all powerful", but is instead, "is all powerful". The same is true of omniscience, and so on.
After the age of the Upanishads, Hinduism gradually departed from this non-dualistic view, and now is functionally (for most Hindus) dualistic, where God is worshipped devotionally as another entity, as in other monotheistic religions. For most, non-dualism is unnatural, and unintuitive, precisely because when it is scrutinized, it collapses into an incoherent system. Yet, it is the oldest and most authentic view of reality in philosophical Hinduism.
The Samhita portions of the Vedas, at base, are religious manuals of sorts composed by a select priest class in ancient India. They contain very little which would be relevant to a modern person. For instance, the Rig Veda consists mostly of hymns to various gods, some of which remain popular in modern Hinduism, with others being mostly supplanted in favor of a more popular deity like Vishnu or Shiva. The Atharva Veda contains what can be called spells or incantations, including for things like success in gambling. Each Veda functioned as a ritual manual for the priest class which guarded the texts, and would be paid by devotees and those in power for their services.
The Vedas were never for everyone to read. Throughout various stages in Hindu history, if a woman, or a member of a certain Caste read or heard them, they would be severely punished. For instance:
Brahma Purana - Chapter 64
19 In regard to the people of the first three castes the rite of holy bath and the repetition of Mantras is in the manner mentioned in the Vedas. The utterance of the Vedic passage is to be avoided by women as well as Śūdras.
Narada Purana - Book 1, Chapter 15
100 I shall mention the fruits of sins accruing to those who engage themselves in the recitation and study of the Vedas in the vicinity of women and Sudras. Listen to it attentively.
101 They are compelled to stand with their heads down and legs upwards. Thus, they are nailed to two pillars and are compelled to inhale smoke continuously in this posture. They stay thus for the period of year of god Brahma (3 trillion years).
Devi Bhagavata Purana - Book 1, Chapter 3
21 The more so because women, Śudras, and the lower Dvijas are not entitled to hear the Vedas; for their good, the Purāṇas have been composed.
Gautama Dharmasutra - Chapter 12
4 Now if (A Śūdra) listens intentionally to (a recitation of) the Veda, his ears shall be filled with (molten) tin or lac.
5 If he recites (Vedic texts), his tongue shall be cut out.
6 If he remembers them, his body shall be split in twain.
The above demonstrate the kind of gatekeeping that these texts were subject to since their inception. They are essentially a religious "trade secret" - magic words used by those favored by society to maintain their status in the religion of Brahminism (the religion of the Brahmins/priests). Later, Hinduism became more democratized, philosophical, and devotional. Originally, it was not so.
These ritual texts, however, were what gave way to works like the Upanishads, the Puranas, and so on, and so they are maintained as foundational by most Hindus. But they do not deserve such a status, and most Hindus do not read them at all.
Additionally, it is known from earlier works that in the past, there were various recensions (Shakhas) of each Veda - sometimes numbering in the dozens for a single Veda - most of which are now totally lost, making having confidence in claiming to have the "original text" of any particular Veda functionally impossible. As a digression, it is also known that a large portion of the Brahmanas, which are considered Shruti, are now lost, and the same is true for many individual Upanishads - later Vedic texts.
The oldest layer, therefore, of the religion of Hinduism, can be summarized as a ritual manual for a small, powerful priest class, which is almost entirely unused by modern Hindus, and which has a dubious textual basis which is nowhere near as robust as it was when the ritual manuals were in active use in ancient India. It contains dozens of hymns describing rituals which no one carries out anymore (the Horse Sacrifice, for instance), and hundreds of references to a drink whose formulation is now lost, and can only be speculated on (Soma).
Therefore, the Vedas are a poor foundation for a religion, or a worldview. Their origin as a money-making/power-preserving enterprise by a privileged class in ancient India is relatively easy to ascertain.
As already mentioned in the above sections, the sacrificial Hinduism of the early Vedic period, and the non-dualistic theology of early philosophical Hinduism, have largely been abandoned in modern Hinduism in favor of a devotional Hinduism which first became popular in the age of the Puranas. However, modern Hinduism is not even a faithful representation of that era of Hinduism, instead rejecting many of the beliefs common in the Puranic age through the Middle Ages.
First, it should be noted that the Puranas are intensely sectarian texts, which contradict each other, and sometimes actively argue against each other. For instance, the Shiva Purana teaches that Vishnu got tricked by Brahma (Shiva Purana 1.7.26-33), and that Shiva is the supreme Brahman, whereas many other Puranas teach that Shiva and Vishnu are the same, or that Vishnu is the supreme Brahman. The Padma Purana teaches that the Matsya, Kurma, Linga, Shiva, Skanda, and Agni Puranas are vicious, and lead one to Hell (Padma Purana 6.236.18-22). So, once one reaches the age of the Puranas, there is no longer a single "Hinduism" to define or examine, but instead, multiple competing sects of Hinduism with incompatible beliefs.
During this period, the view of the religion of Hinduism toward widows was extremely negative. In the first place, it was extremely common to teach that marrying a widow was forbidden, and disgraceful:
Mahabharata - Book 1, Sambhava Parva, Chapter 104
31 At these words of his wife, Dirghatamas said, I lay down from this day as a rule that every woman shall have to adhere to one husband for her life. Be the husband dead or alive, it shall not be lawful for a woman to have connection with another. And she who may have such connection shall certainly be regarded as fallen.
Vamana Purana - Chapter 12
1 Sukesi said, What action leads to these hells and in what manner? Please tell me this, O venerable Brahmanas. Great is my curiosity about it.
2 The sages said: O Sukesi, listen from us the action by the performance of which one goes to the hells to undergo the suffering caused by such actions.
35 Those who remarry widows and those that defile unmarried girls, and the ancestors of the offspring of such unions are made to eat worms and ants.
Padma Purana - Book 3, Chapter 56
14 (Vyāsa said) (He should) especially (avoid) the food of a widow who is remarried, so also of the husband of a woman who is married twice.
Kurma Purana - Book 2, Chapter 17
14 One should particularly avoid the food served by a widowed woman who has remarried, nor should he partake of the food served by one who carnally enjoys the wife of his dead brother. Similarly, he should avoid partaking of the food served with contempt or anger, spurning, dislike or arrogance or astonishment.
Skanda Purana - Book 3, Section 2, Chapter 21
15 (Yājñavalkya says) It is mentioned by Kāśyapa that these burn the family like fire: The daughter of a widow who marries again should be always avoided. She is the lowliest born.
16 One who has been verbally given in marriage, one mentally betrothed, one whose marital sacred string tying is over, one who is offered by touching water or taken by the hand,
17 One who has gone round the fire and one who is a child of a widow remarried - all these burn the family like fires (says Kāśyapa).
Additionally, during this period, the religion of Hinduism taught that it was virtuous for a widow to burn herself alive on her husband's funeral pyre, a practice known as Sati:
Vishnu Smriti - Chapter 25
1 Now the duties of a woman (are as follows):
12 Not to act by herself in any matter;
13 To remain subject, in her infancy, to her father; in her youth, to her husband; and in her old age, to her sons.
14 After the death of her husband, to preserve her chastity, or to ascend the pile after him.
Parashara Smriti - Chapter 4
30 If a woman follows her departed lord, by burning herself on the same funeral pile, she will dwell in heaven for as many years as there are hairs on the human frame, - which reach the number of three crores and a half (35 million).
Daksha Smriti - Chapter 4
19 A woman, who, after the demise of her husband, ascends the funeral pyre, becomes of good conduct, and lives gloriously in the celestial region.
Agni Purana - Chapter 222
21 A widow as well as the wife of a man who has gone abroad, should not decorate her person. She should always be bent on the worship of god and the welfare of the husband.
22 (A woman whose husband has gone abroad) should wear some ornaments for the welfare (of the husband). The woman (widow) who enters the (funeral) fire along with the (dead) husband would also reach heaven.
Kurma Purana - Chapter 34
108 A woman who enters the funeral pyre along with her husband, shall uplift him even if he is a Brahmana-slayer, an ungrateful fellow or one defiled by great sins.
109 Learned men know this to be the greatest expiation for women.
Garuda Purana - Chapter 107
29 A wife immolating herself with the dead body of her husband at the same funeral pile, shall live in heaven for as many number of years as there are hairs on the human body. A man, bitten by a dog, or by an animal of the kindred species, is purified by mentally reciting the Gayatri Mantra.
Skanda Purana - Book 3, Section 2, Chapter 7
53 A chaste woman who follows her husband from the house to the cremation ground for self-immolation undoubtedly attains the merit of a horse-sacrifice at every step.
54 Just as a snake-catcher forcibly retrieves a snake from its hole, so also a chaste woman retrieves her husband from the messengers of Yama and goes to heaven.
Modern Hindus would, for the most part, (rightly) reject these extremely discriminatory teachings regarding widowed women. But, that is a departure from the Hinduism taught in the age of the Puranas.
Additionally, as covered in a section above, the Puranas commonly teach that women and certain Castes cannot read or listen to the Vedas. These additional quotes demonstrate that this was an extremely common teaching in Puranic Hinduism:
Bhagavata Purana - Book 1, Chapter 4
25 The three Vedas are not to be heard by women, Śūdras and lowest among the twice-born ones (i.e. unworthy persons of Brāhmaṇa, Kṣatriya, and Vaiśya castes). In order that the welfare of such (persons) ignorant about actions leading to the good (results) may be (achieved) this way here, that the sage (Vyāsa) compiled the narrative called the Bhārata.
Brahma Sutras - Book 1, Chapter 3
38 And on account of the prohibition in Smriti of (the Sudras) hearing, studying and understanding (the Veda) and performing Vedic rites (they are not entitled to the knowledge of Brahman).
Manusmriti - Chapter 4
98 After this, (the Brāhmaṇa) shall diligently read the Vedas during the bright fortnights, and all the subsidiary sciences during the dark fortnights.
99 He shall recite, not indistinctly, nor in the proximity of Śūdras; nor shall he go to sleep again, at the end of night, when he is tired after having recited the Veda.
Manusmriti - Chapter 9
17 Manu assigned to women sleep, sitting, ornament, lust, anger, dishonesty, malice and bad conduct.
18 For women there is no dealing with the sacred texts; such is the rule of law; the fact is that, being destitute of organs and devoid of sacred texts, women are 'false'.
Vishnu Smriti - Chapter 30
3 During the period (subsequent upon the ceremony of Upakarman and) intermediate between it and the ceremony of Utsarga, the student must read the Vedangas
10 (He must not study) in a village in which a corpse lies
11 Nor during a battle
12 Nor while dogs are barking, jackals yelling, or donkeys braying
13 Nor while the sound of a musical instrument is being heard
14 Nor while Sudras or outcasts are near
Modern Hinduism as it is popularly practiced typically would not make such declarations, or try to enforce such rules, because it is much more egalitarian.
The Hinduism of the Puranas also taught a rigid and discriminatory Caste System, where Brahmins are superior to other human beings in many ways, and those who are born into poor circumstances, low Castes, or with birth defects, are being punished for their sins in a past life.
For instance:
Garuda Purana - Chapter 5
1 Garuḍa said: Tell me, O Keśava (Vishnu), by what sins particular signs are produced, and to what sorts of birth such sins lead?
2 The Blessed Lord said: The sins on account of which the sinful returning from hell come to particular births, and the signs produced by particular sins - these hear from me.
3 The murderer of a Brāhmaṇ becomes consumptive, the killer of a cow becomes hump-backed and imbecile, the murderer of a virgin becomes leprous - all three born as outcastes.
4 The slayer of a woman and the destroyer of embryos becomes a savage full of diseases; who commits illicit intercourse, a eunuch; who goes with his teacher's wife, diseased-skinned.
5 The eater of flesh becomes very red; the drinker of intoxicants, one with discoloured teeth; the Brāhmaṇ who, on account of greed, eats what should not be eaten, becomes big-bellied.
6 He who eats sweet foods, without giving to others, becomes swollen-necked; who gives impure food at a Śrāddha ceremony is born a spotted leper.
7 The man who, through pride, insults his teacher, becomes an epileptic; who despises the Vedas and the Śāstras certainly becomes jaundiced.
8 Who bears false witness becomes dumb; who breaks the meal-row becomes one-eyed; who interferes with marriage becomes lipless; who steals a book - is born blind.
9 Who strikes a cow or a Brāhmaṇ with his foot is born lame and deformed; who speaks lies becomes a stammerer, and who listens to them becomes deaf.
10 A poisoner becomes insane; an incendiary becomes bald; who sells flesh becomes unlucky; who eats fled of other beings becomes diseased.
11 Who steals jewels is born in a low caste; who steals gold gets diseased nails; who steals any metal becomes poverty-stricken.
Manusmriti - Chapter 8
380 Let him never slay a Brahmana, though he have committed all (possible) crimes; let him banish such an (offender), leaving all his property (to him) and (his body) unhurt.
381 No greater crime is known on earth than slaying a Brahmana; a king, therefore, must not even conceive in his mind the thought of killing a Brahmana.
The above are just two examples of many hundreds of texts from this era which teach that there is a class of superior human beings who find themselves in power and authority because of their own virtue in a past life. That was the religion of Puranic Hinduism. If a modern Hindu rejects that a Brahmin is superior to them because of the Brahmin's own virtue, then they reject the Hinduism of the Puranas.
Finally, the Puranas teach a vast array of cosmological and scientific doctrines which most educated Hindus would be embarrassed of, and would reject.
For instance, multiple Puranas teach that one could select the gender of their child by having intercourse only on certain nights of their menstrual cycle:
Brahma Purana - Chapter 113
75 In the case of the people of all castes, a woman in her monthly course should be avoided for the first four nights. To prevent the birth of a female child, one shall avoid the fifth night also.
76 Thereafter he shall carnally approach her on the sixth night or other excellent nights even in number. Boys will be born (if intercourse is carried out) on even nights and girls on nights odd in number.
Some Puranas teach that the direction one sleeps affects their health:
Vishnu Purana - Book 3, Chapter 11
After eating his evening meal, and having washed his feet, the householder is to go to rest. His bed is to be entire, and made of wood: it is not to be scanty, nor cracked, nor uneven, nor dirty, nor infested by insects, nor without a bedding: and he is to sleep with his head either to the east or to the south; any other position is unhealthy. In due season a man should approach his wife, when a fortunate asterism prevails, in an auspicious moment, and on even nights, if she is not unbathed, sick, unwell, averse, angry, pregnant, hungry, or over-fed.
Multiple Puranas teach that the moon is larger than the sun, and are also incorrect about how large the sun and moon are:
Brahmanda Purana - Book 2, Chapter 21
7 The extent of the disc of the sun is nine thousand Yojanas. Its girth is three times the extent.
8 In diameter as well as girth (Circumference), the moon is twice as much as the sun.
Many Puranas present a water cycle which involves the sun evaporating water, then communicating that water to the moon, which then communicates that water to the earth:
Brahmanda Purana - Book 2, Chapter 22
13 In the course of his revolution, O Brāhmaṇas, he illuminates the quarters. With the mass of his rays and accompanied by the wind all round, the sun takes away the water of the entire world, O excellent Brāhmaṇas.
14 The moon transmits the entire water drunk by the sun. The torrential current of water (Downpour of rain) functions through Nāḍis (veinlike vessels) accompanied by the wind.
15 The water that oozes out of the moon remains in the food (i.e. plants and food crops).
16 Being obstructed by the wind, the clouds scatter water over the earth. Thus water is continuously raised up (i.e. turned into vapour) and dropped down.
Many Puranas teach that the components of the human body develop out of one another, which scientifically speaking, is false:
Vayu Purana - Book 2, Chapter 35
44 Rasa (the lymphatic juice) becomes blood; flesh is developed out of blood.
45 From flesh fat is produced, and bone is developed from fat. Pith and marrow are developed from bone, and semen virile is generated from marrow;
Most modern Hindus who are educated will reject all of these things. Therefore, they are rejecting the doctrines of Puranic Hinduism.
In summary, the religion of Hinduism in modern times is fundamentally different from the sacrificial religion of the early Vedas, has largely abandoned the non-dual metaphysics of the Upanishads, and rightly rejects many of the morally abhorrent and unscientific doctrines of medieval, Puranic Hinduism. What remains is not a continuous religious tradition, but a modern reconstruction that selectively borrows from multiple incompatible historical systems.
Hinduism is nonsense because its foundation is rooted in a privileged class of priests in ancient India who utilized people's superstitions to gain wealth and power for themselves. The first layer of philosophical Hinduism teaches an incoherent view of reality. And, the Hinduism of the age of the Puranas is indefensible, even being rejected by most modern Hindus.