Thomas Jech, Set Theory (Millenium edition): Contains everything (and more). Work through it cover to cover and you're a set theorist!
Kenneth Kunen, Set Theory: An introduction to independence proofs: This is the most elegant and rigorous introduction to independence proofs.
Kechris, Classical Descriptive Set Theory: A very well written introduction to descriptive set theory, containing most of the results in the field which you are going to use frequently.
Tomek Bartoszynski and Haim Judah, Set Theory: On the structure of the real line: This is perhaps the best book about set theoretic analysis of the real line, special sets of real numbers, and forcing notions adding new reals. Fascinating.