The best introduction to set theory

In my opinion, Yiannis N. Moschovakis's Notes on Set Theory is the best introduction to this topic for several reasons. First of all, it's written in a very clear and understandable style (which is far from usual even in undergraduate math books). Secondly, Moschovakis doesn't begin with a list of all ZFC axioms (as it is very often done), but begins developing his theory with as few axioms as possible and thereby is able to show which axioms are really needed for what and how. Last but not least, he manages to include lots of interesting topics such as recursion theory, order theory or descriptive set theory, which usually are omitted in introductions to set theory. There is even a really nice construction of the real numbers. By the way: for those of you who want to know even more about set theory, there's still the bible.

If you're looking for a general and very well readable overview over mathematical logic, I recommend a look at A Tour Through Mathematical Logic by Robert S. Wolf.