Perseus is a massive digital library, whose “flagship collection, under development since 1987, covers the history, literature and culture of the Greco-Roman world.” You can read & search in English, Greek, & Latin. It’s the online academic gold-standard.
This is the easiest place to access primary sources for NT studies, (from the 1st-3rd centuries C.E.). Covering the NT, Apocrypha, Gnostic Lit, Church Fathers, sources are arranged by date and by type.
“Early Jewish Writings is the most complete collection of Jewish documents from antiquity with translations, introductions, and links.” This is Peter Kirby’s easy-to-navigate collections of Josephus, Philo, the Pseudepigrapha, Deuterocanon.
CCEL hosts “hundreds of classic Christian books selected for edification and education.” While the earliest texts on the site are the Epistle of Barnabas & Irenaeus’ letters, the site consists mostly of authors from the 1100’s onwards.
“An online resource dedicated to the Gospel of Thomas other early [non-canonical] Christian Gospels.” Not only does the site give you information, but provides links to online texts for each gospel.
Soncino Babylonian TalmudTranslated by Rabbi Dr. I. Epstein, this contains most of the Babylonian Talmud.
Guide to Medieval Christian SpiritualityThis site contains both primary and secondary sources on Medieval Christian Spirituality, and is an excellent place to begin your research.
There are many good places to start research on the DSS. This site, hosted at ibiblio.org, is part of an exhibition at the Library of Congress in Washington DC. It’s a very good primer that equips you for further research.
Advanced Papyrological Information System (APIS)This site provides online access to the papyrological collection at the University of Michigan as well as collections from other universities such as Columbia, Duke, and Yale.