There are several tactics for tracking down the full reference to a quotation. It is easy enough to put a quotation into Google and find out to whom it has been attributed. It is often much harder to find a source for the original quotation, especially when the quotation has been paraphrased, poorly cited, or mis-attributed.
For more widely known quotes, our books of quotations can be invaluable. From the general and classic Bartlett's Familiar Quotations to the more specific Oxford Dictionary of Humorous Quotations, they are routinely indexed by author and by subject. These books have different areas of overlap, so be sure to browse the reference section for multiple sources if you don't find your quote in the first book you try.
The subject index is particularly useful when you know a paraphrased quote or are not sure of the author:
"I wouldn't want to be in a club that would have me as a member." -perhaps Mark Twain?
A search in Bartlett's for club as the subject leads us directly to the actual quote:
"Please accept my resignation. I don't care to belong to any club that will accept me as a member."
-Groucho Marx, The Groucho Letters (1967)
We also have online access to several books of this type. Using Credo Reference allows you to search seven quotation books at once, while Oxford Reference allows you to search three others. Non-Williams users might want to try Quotations at Bartleby.com, a useful and free resource.
These can be the most difficult to deal with, especially since variations in translation may produce several versions in English of the original. Consider the last line of Albert Camus' The Myth of Sisyphus, which appears as "One must believe that Sisyphus is happy" in Bartlett's and as "One must imagine Sisyphus happy" in another translation.
For cases such as these, it makes sense to turn to the original language, in this case French. The Oxford Dictionary of Quotations gives an English translation and the French original, "Il faut imaginer Sisyphe heureux." The Dictionnaire des Citations Françaises agrees, and even with limited French one can find Sisyphe in the subject index and thereby the original quote.
Quotes that are not cataloged in the works described above can still be discovered with a little sleuthing. For this sort of searching, our full text resources are quite helpful. You may find the actual quote and citation online, or a decent reference to a paper resource.
Google Book Search is excellent for finding lines from plays and citations for quotations. Even if you can't see all the material online, you will have a reference to a book that can then be tracked down in the library for verification. Don't use quotation marks when you search this way if you think the quote might have been paraphrased - that way Google will return near matches as well as exact ones.
You might use a news database like ProQuest Historical Newspapers (US) to find a quote from an interview.
A full text scholarly resource such as JSTOR will often turn up references to an obscure quote.
If you think the quote might be a line from a poem, you can use one of our poetry indexes, which index keywords, titles, and first and last lines of poems.
Finally, if you have an inkling of the author, concordances can be used. Concordances are a list of the major words used in a work, cross referenced to the passages in which they are used. While most often used for scriptural references, we also have concordances for prolific authors such as Shakespeare, Milton, Chaucer, etc. Try a search for your author and the word "concordances" (plural) in FRANCIS to find concordances for other authors.
If all else fails, ask a librarian for help.