What are good books/other readings for elementary set theory?
I am going to go out on a limb and recommend a more elementary book than (I think) any of the ones others have mentioned.
I claim that as a pure mathematician who is not a set theorist, all the set theory I have ever needed to know I learned from Irving Kaplansky's Set Theory and Metric Spaces. (And, you know, I also enjoyed the part about metric spaces). Kaplansky spent most of his career at the University of Chicago. Although he had left for MSRI by the time I got there in the mid 1990's, nevertheless his text was still used for the one undergraduate set theory course they offered there. (Not that I actually took that course, but I digress...)
In fact I think that if you work through this book carefully -- it's beautifully written and reads easily, but is not always as innocuous as it appears -- you will actually come out with more set theory than the average pure mathematician knows.
Apologies if you actually do need or want to know some more serious stuff: there's nothing about, say, cofinalities in there, let alone forcing and whatever else comes later on. But maybe this answer will be appropriate for someone else, if not for you.
Added: I suppose I might as well mention my own lecture notes, available online here (scroll down to Set Theory). I think it is fair to say that these are a digest version of Kaplansky's book, even though they were for the most part not written with that book in hand. [However, last week David Speyer emailed me to kindly point out that I had completely screwed up (not his words!) one of the proofs. He also suggested the correct fix, but I didn't feel sanguine about it until I went back to Kaplansky to see how he did it.]
The description All the set theory I have ever needed to know on the main page is not meant to be offensive to set theorists (and I hope it isn't) but rather an honest admission: here is the little bit of material that goes a very long way indeed. Note especially the word need: this is not to say that these 40 pages contain all the set theory I want to know. For instance, I own Cohen's book on forcing and the Continuum Hypothesis, and I would certainly like to know how that stuff goes...
[Come to think of it: I would be highly amused and interested to read 40 pages of notes entitled All the number theory I have ever needed to know written by one of the several eminent set theorists / logicians who frequent this site and MO. What would make the cut?]
I recommend Naive Set Theory by Halmos. It's a friendly, thin and fun to read introduction to set theory.
http://books.google.com/books?id=x6cZBQ9qtgoC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
Enderton's book should be a gentle, easy read for an undergraduate
http://www.amazon.com/Elements-Set-Theory-Herbert-Enderton/dp/0122384407/ref=pd_sim_b_26