For Islam, the primary sources are the Qur’an, the collections of “hadith” (Islamic traditions), other traditional Islamic texts, and modern tracts, books, and websites by Muslims.
The secondary sources are, ideally, scholarly sources by people with Ph.D’s in the study of Islam. A bibliography of these can be found
here.
Good advice from Contarini. For a good translation of the Qur’an, I would recommend starting with the one by Muhammad Asad,
The Message of the Qur’an (Gibraltar: Dar al-Andalus, 1980). It has a good commentary with it which is very useful. You can download a pdf version of it here:
muhammad-asad.com/Message-of-Quran.pdf
A good general introduction to the Qur’an, in my opinion, is the one by the South African Muslim scholar Farid Esack,
The Qur’an: A Short Introduction (Oxford, England: Oneworld, 2002).
If you want to read the earliest and most important Muslim biography of Muhammad, there is the
Sirat Rasul Allah by Ibn Ishaq who died in 767 AD. His biography or sira has been translated by Alfred Guillaume, *The Life of Muhammad: A Translation of Ibn Ishaq’s Sirat Rasul Allah *(London: Oxford University Press, 1955). It is still in print.
For a biography of Muhammad based on many original sources, try Martin Lings,
Muhammad: his life based on the earliest sources (New York: Inner Traditions
International, 1983). Martin Lings, now deceased, had a PhD from the School of Oriental and African Studies in London and was Keeper of Oriental Printed Books and Manuscripts at the British Library.
If you want to read selections from some original commentaries (i.e. tafasir) on the Qur’an, there is Mahmoud Ayoub.
The Qur’an and Its Interpreters, Vols. 1-2. (Albany, NY: State University of New York Press, 1984). For the first two surahs of the Qur’an (al-Baqara & Al 'Imran), it has selections of commentary (called tafsir in Arabic) for each verse from numerous scholars all the way from Tabari (who died in 923 AD) to Sayyid Qutb (who died in 1966), as well as selections from a well known Shi’i commentator (Tabataba’i) and a Sufi commentator (Ibn 'Arabi) and many others besides.
A good general introduction to Islam is Sachiko Murata and William C. Chittick,
The Vision of Islam (New York: Paragon House, 1994).
If you want to learn about Sufism or mystical Islam, any book by Professor William C. Chittick is a good choice. He is a very good writer and explains complicated material very well (anything about the thought of the famous Sufi Ibn 'Arabi is complicated). I would recommend
Sufism: A Short Introduction (Oxford, England: Oneworld, 2000), or
The Sufi Path of Knowledge: Ibn al-‘Arabī’s Metaphysics of Imagination (Albany, NY: State University of New York Press, 1989) or
The Sufi Path of Love: The Spiritual Teachings of Rumi (Albany, NY: State University of New York Press, 1983).
Another good general introduction to Sufism is Annemarie Schimmel,
Mystical Dimensions of Islam (Chapel Hill, NC: University of North Carolina Press, 1975). She was a professor at Harvard University.
Probably the best general introduction to Twelver Shi’ism (the largest branch of Shi’i Islam) is Moojan Momen,
An Introduction to Shi‘i Islam: The History and Doctrines of Twelver Shi‘ism (New Haven, CT: Yale University Press, 1985).
A good introduction to hadith which are collections of sayings by or about the Prophet Muhammad, I would recommend, Jonathan A. C. Brown, *Hadith: Muhammad’s Legacy in the Medieval and Modern World *(Oneworld, 2009). Brown is a convert to Islam by the way and an associate professor at Georgetown University.