The stories we read in class can be found here, in PDF format: Arabian Nights RTI Readings
This site has a collection of various translations, including Richard Burton's and Andrew Lang's (look down the red sidebar). I was able to help a little tiny bit with the Burton translation; you can find my name on the site if you look carefully!
Here's Andrew Lang's translation at The Internet Sacred Texts Archive.
Most of the texts we read appear in Malcom Lyons' translation, published by Penguin Classics. (Amazon.com)
Another good contemporary translation (if you're looking for a book) is Husain Haddawy's (Amazon.com), though it only contains about 35 tales, I believe.