The most awesome site for Fairytales is http://www.surlalunefairytales.com/ which does exactly what it says, providing texts, whole e-books, discussion forums and comparative studies of tale-types.
Also excellent is http://www.pitt.edu/~dash/folktexts.html, D.L. Ashliman’s site hosted obligingly by the University of Pittsburgh, even though he hasn’t worked there since 2000 – you can google his homepage if you want to contact him.
Andrew Lang’s series of Fairy Books have provided me with many tales, and especially unusual versions of them from the East, or from Eastern Europe. This link http://www.mythfolklore.net/andrewlang/ obligingly gives you a “Fairytale of the Day” just to start you off, so you can avoid the hassle of choosing, but also links to all the books.
Joseph Jacobs compiled several collections which can be found here http://www.gutenberg.org/ebooks/author/2470
Other sources appear on relevant pages, but I will update this page as I find new collections.
Here are some specific sources for tales of particular kinds:
http://www.sacred-texts.com/neu/roma/gft/index.htm Tales told by the Roma and Sinti in Eastern Europe
http://oaks.nvg.org/danish-folktales.html As it says, Danish folktales
http://www.sacred-texts.com/neu/basque/bl/ Basque legends – I tell Mahistruba, the Master Mariner
http://www.worldoftales.com/ has a fair range, which overlaps with some other sites
http://www.wollamshram.ca/1001/Pentamerone/pentamerone1_main.htm gives Burton’s translation of the Pentamerone, and http://www.wollamshram.ca/1001/Vikram/vikram.htm gives Burton’s wilful adaptation of the Baital Pachchisi, or The King and the Corpse, which is much more accurately rendered here, and lower down the page are further links to all kinds of riches, including the Thousand and One Days of Muhli, and countless other examples of Persian literature at http://persian.packhum.org/persian/main
And some stories I have acquired from other storytellers, particularly Taffy Thomas, Daniel Morden, Hugh Lupton, Red Phoenix, Christobel Thomas and Giles Abbott, though I naturally tell them my way.
I have just stumbled on a fascinating site run by Laura Gibbs, http://mythfolklore.blogspot.co.uk/, and I shall no doubt be browsing and pillaging it for the next few months.