Hinduism

The Puranas are post-Vedic texts which typically contain a complete narrative of the history of the Universe from creation to destruction, genealogies of the kings, heroes and demigods, and descriptions of Hindu cosmology and geography. There are 17 or 18 canonical Puranas, divided into three categories, each named after a deity: Brahma, Vishnu and Shiva. There are also many other works termed Purana, known as 'Upapuranas.'

The Vishnu Purana by H.H. Wilson [1840]
This is a primary text of the Vaishnava branch of Hinduism, and one of the canonical Puranas of the Vishnu category. Among the portions of interest are a cycle of legends of the boyhood deeds of Krishna and Rama. H.H. Wilson was one of the first Europeans to translate a Hindu sacred text from the original Sanskrit. His style and annotations are exceptional and very readable.

Other Primary Texts

The Laws of Manu George Bühler, translator [1886] (Sacred Books of the East, vol. 25)
Manu was the legendary first man, the Adam of the Hindus. This is a collection of laws attributed to Manu.
The Sacred Laws of the Âryas, Part I (SBE 2) George Bühler translator [1879] (Sacred Books of the East, Vol. 2)
These are Hindu law books written by the sages Âpastamba and Gautama, in the first millenium B.C.
The Sacred Laws of the Âryas, Part II (SBE 14) George Bühler translator [1879] (Sacred Books of the East, Vol. 14)
These are Hindu law books written by the sages Vasishtha and Baudhâyana, in the first millenium B.C.
The Institutes of Vishnu (SBE 7) Julius Jolly, translator [1880] (Sacred Books of the East, Vol. 7) This is also one of the law books of Hinduism. It contains several notable passages, including descriptions of yogic practises, and a moving hymn to the Goddess Prajapati.
The Satapatha Brahmana
A primary source for Vedic-era mythology, philosophy and magical practices. The complete five part Sacred Books of the East Satapatha Brahmana translation is now online:
  Part I (SBE12)
  Part II (SBE26)
  Part III (SBE41)
  Part IV (SBE43)
  Part V (SBE44)

The Epics

The Mahabharata and Ramayana are the national epics of India. They are probably the longest poems in any language. The Mahabharata, attributed to the sage Vyasa, was written down from 540 to 300 B.C. The Mahabharata tells the legends of the Bharatas, a Vedic Aryan group. The Ramayana, attributed to the poet Valmiki, was written down during the first century A.D., although it is based on oral traditions that go back six or seven centuries earlier. The Ramayana is a moving love story with moral and spiritual themes that has deep appeal in India to this day.

In addition, a key Hindu sacred text, the Bhagavad Gita, is embedded in Book Six of the Mahabharata.

Mahabharata

The Mahabharata now has its own page:

The Mahabharata in Sanskrit
Unicode text with parallel Devanagari and Romanization.

The Ramayana

Rámáyan Of Válmíki ,translated by Ralph T. H. Griffith [1870-1874]
This is the first complete public domain translation of the Ramayana to be placed online.

The Ramayana in Sanskrit
Unicode text with parallel Devanagari and Romanization.

Abridged Versions

The Ramayana and Mahabharata R. Dutt translator [1899]
A very readable abridged version of these epics.

Indian Idylls Sir Edwin Arnold, translator [1883] 279,713 bytes
More stories from the Mahabharata, rendered in poetry.

Bhagavad Gita

The Bhagavad Gita, usually considered part of the sixth book of the Mahabharata (dating from about 400 or 300 B.C.), is a central text of Hinduism, a philosphical dialog between the god Krishna and the warrior Arjuna. This is one of the most popular and accessible of all Hindu scriptures, required reading for anyone interested in Hinduism. The Gita discusses selflessness, duty, devotion, and meditation, integrating many different threads of Hindu philosophy.

The Bhagavadgîtâ (SBE 8) with the Sanatsugâtîya and the Anugîtâ translated by Kâshinâth Trimbak Telang, (Sacred Books of the East, Vol. 8) [1882]
This is a scholarly prose translation of the Bhagavad Gita with two other similar, less well known, works from the Mahabharata.

The Bhagavad Gita in Sanskrit
A Romanized Unicode version.

The Bhagavad Gita
A modern prose translation, sanctioned by the International Gita Society.

Vedanta

The Vedântâ-Sûtras (SBE 48) with commentary by Râmânuja, translated by George Thibaut; (Sacred Books of the East, Vol. 48) [1904]
The Vedântâ-Sûtras Part I (SBE 34) with commentary by Sankarâkârya, translated by George Thibaut; (Sacred Books of the East, Vol. 34) [1890]
The Vedântâ-Sûtras Part II (SBE 38), with commentary by Sankarâkârya, translated by George Thibaut; (Sacred Books of the East, Vol. 38) [1896]
The Crest-Jewel of Wisdom and other writings of Śankarâchârya; translation and commentaries by Charles Johnston [1946, copyright not renewed]

Later Texts

The Yoga Sutras of Patanjali 80,965 bytes This concise work describes an early stage in the philosophy and practise of Yoga. Dating from about 150 B.C., the work shows dualist and Buddhist influences. The Yoga Sutras are required reading if you are interested in Yoga and meditation.
The Sánkhya Aphorisms of Kapila
translated by James R. Ballantyne [1885]
Kalidasa: Translations of Shakuntala and Other Works by Arthur W. Ryder [1914]
Kalidasa, who lived in the fifth century C.E., is known as 'the Shakespeare of India.' The drama of Shakuntala is his masterpiece.
Verses of Vemana
Vemana was a 17th century South Indian poet. Translated from the Telugu by C.P. Brown [1829]
Black Marigolds
Translated by Edward Powys Mathers [1919] A free verse translation of the Caurapañcāśikā of Bilhana, an 11th century Kashmiri poet.
Vikram and the Vampire Sir Richard Burton, translator. [1870]

The Vimanika Shastra
Translated by G.R. Josyer [1973]

Modern Books

Old Deccan Days by Mary Frere [1868]
Râmakrishna, His Life and Sayings by F. Max Müller [1898]
How To Be A Yogi by Swâmi Abhedânanda [1902]
Twenty-two Goblins by Arthur W. Ryder [1912]
Indian Fairy Tales by Joseph Jacobs [1912]
Indian Myth and Legend by Donald A. Mackenzie [1913]
Hindu mythology from the earliest times through the the Mahabharata and Rayamaya.
Hindu Mysticism by S.N. Dasgupta [1927, not renewed]

Works of Sister Nivedita (Margaret E. Noble)

Kali the Mother by Sister Nivedita (Margaret E. Noble) [1900]
The Web of Indian Life by Sister Nivedita (Margaret E. Noble) [1904]
Studies from an Eastern Home by Sister Nivedita (Margaret E. Noble) [1913]

Works of Rabindranath Tagore

Gitanjali [1913]
Saddhana, The Realisation of Life [1915]
The Crescent Moon [1916]
Fruit-Gathering [1916]
Stray Birds [1916]
The Home and the World [1915]
Thought Relics [1921]
Songs of Kabîr