The other books mentioned here are also great, but I think Cat's Cradle is the best book to start with.
Mother Night is one that gets less attention, but it's a hell of a book.
I second this. I've read a lot of his books, with Breakfast of Champions being my current favorite, but I'm halfway through Mother Night and I gotta say it's my new favorite.
It might not be the best place, but an easy entry is Welcome to the Monkey House. It's a collection of 25 short stories that were published in magazines in the '50s and '60s.
Slaughterhouse Five is probably his most famous, but I think Breakfast of Champions is his best.