The best book I've seen on set theory for the serious student is Yiannis Moschovakis' Notes on Set Theory . Deep, broad and beautifully written with lots of good exercises. It's the book I wish I'd known about when I was taking Russell Miller's set theory course at CUNY. The old standby which I actually did use for the course is Herbert Enderton's Elements of Set Theory. This book is also beautifully written, but is deeper in a number of respects and Enderton is sometimes confusing since he isn't clear sometimes on the distinction between a set, it's subsets and it's elements.
(To be honest, though, this is not so much a failing of Enderton's book, but of axiomatic set theory in general when it is presented in natural language rather then mathematical logical language. But that's the great chicken-or-the-egg debate of axiomatic set theory: logic first or no logic?)
There's a few others,but those are my faves.