Start with Reference Works One of the best places to start your research is in reference works: in particular, the Oxford Classical Dictionary, (available online). Brill's New Pauly (still incomplete in English) is also useful. Most entries include brief bibliographies that will direct you to important work on the topic. Also useful are the essays included in the series of Companions published by Cambridge University Press (available online) and Blackwell, as well as the relevant chapters of the Cambridge Ancient History, which also provides extensive bibliographic references both to ancient sources and recent scholarship.
Track down references. The works you find will include footnotes and/or bibliographies that will lead you to further resources on your topic. You can see if a particular journal article is available online using Citation Linker. Note, however, that most scholarly material in classical studies is not available online. For books, check the title in the online catalog and find more books with the same subject headings and by browsing on nearby shelves.
Use Indexes and Databases. L'Année philologique is the standard index for classical studies, although its coverage begins about three years before the present. More recent coverage can be found using the Gnomon (in German) and TOCs-IN (Tables of Contents of Interest to Classicists) databases. Archäologische Bibliographie (Dyabola) is a very comprehensive index of work in classical archaeology. It is not, however, for the faint of heart. JSTOR is an indispensable full-text collections of scholarly journals. It includes many important journals for ancient history and classical studies. Note however, that, in most cases there is a three to five year delay before journals are added to JSTOR, a "moving wall": it does not contain the most recent scholarship.