Classicists tend to use the Chicago Manual of Style rules for citation, though that depends on what venue the work is intended for--publishers often have their own rules or variations on the standard. Turabian, a derivative of Chicago designed for undergraduate research, is also popular for less formal work.
For more humanities-focused work, the Modern Language Association (MLA) style may be used.
For the most part, citations of ancient works begin with the author's name (sometimes abbreviated), followed by a short or abbreviated title, usually in italics. (In cases where only one work by an author is known, the title may be omitted.) This is followed by a sequence of numbers and/or letters that indicate the specific subdivisions of the work. Arabic numerals tend to favored in recent publications, but older citations often use Latin numerals as well. Exactly how works are subdivided varies. Book, chapter, section and/or line numbers are often provided. Some works are cited by page numbers of standard, or once-standard, editions. Scholarly editions of these texts will be subdivided by these schemes, as will many translations.
Abbreviations should be taken from an authoritative source such as the Oxford Classical Dictionary (see below). You should also provide bibliographic details of the edition and/or translation of the work you are using: generally when the work first appears in your notes and in your bibliography.
For more information see the Chicago Manual of Style, 17th edition (Chicago, 2017) §§14.242–250, also available online. See also The SBL Handbook of Style For Ancient Near Eastern, Biblical, and Early Christian Studies (Peabody, MA, 1999), §8.3.14.
Classical works are typically referred to by chapter and verse, like Scripture, rather than in conventional footnote/reference style. Example:
Plato Protagoras 309c
Virgil Aeneid 2.250-252
Thanks to the Yale University Library Classics guide for much of this information!