Using quotes from plays

My understanding is that there never was copyright on Shakespeare's work (it predates Queen Anne - ie the existence of any form of copyright).

Basically you need to find out if there is still a rights holder, but generally speaking you can be certain of being safe if the author died over a hundred years ago.

Check out Project Gutenberg - http://www.gutenberg.org/

Look up the expression "public domain."

In general, anything published in the US before 1923 is in the public domain. None of Shakespeare's works were ever copyrighted and so they are all in the public domain and can be used without paying any royalties to the author's estate. Here is a handy guide for determining when something passes into the public domain.