N.B. The following was initially an answer to a very similar question, now closed as duplicate. As, imo, my answer suits this question just as well (if not even better), this question is still open and has no answer, and I think that I gave some suggestions that I haven't seen in the possible duplicates, I'm taking the liberty to give it here (with very slight modifications) instead. If this is somehow out of line, let me know and I'll delete it.
First: I am not a set theorist, but I have a BSc in mathematics, and I am almost done with my MSc (also in mathematics), and set theory happens to be one of those subjects outside of what I do, that I find especially fascinating. I have taken a graduate level course in logic with set theory, but I have only looked at the pure course in set theory and forcing. So, my answer is very much from a students perspective, which I hope is a good thing.
When we did set theory as part of a more advanced course in logic (but still only technically requiring a first course in mathematical logic), we mainly used
1.) R. Cori, D. Lascar; Recursion Theory, Gödel's Theorems, Set Theory, Model Theory. Oxford University Press.
This is part II of a series (duology?) of books on logic (first one here). I must say, starting out with axiomatic set theory, I really liked this one, and this is perhaps especially good if you want a somewhat concise yet rigorous introduction. It will also suit you well if you wanted lots of exercises, and it also has solutions to them (which ime is unusual at this level). (The first book is also a very good rigorous introduction to mathematical logic.)
I also really like
2.) Notes on Set Theory, Second edition, Springer 2006, by Y.N. Moschovakis,
which is of course a more complete book on set theory, but includes axiomatics. Both these books are, in my opinion, concise (at least 1), rigorous, yet accessible, but should still be challenging enough.
The main book used in the pure set theory course at our department is
3.) Kenneth Kunen, Set Theory – an Introduction to Independence Proofs, North-Holland 1980
N.B. I have only skimmed this one, however it looks good, and imo, all book recommendations and choices for course literature in logic courses at our department that I've read, has been of very high quality. (We have a very long tradition of logic and many people doing logic related research here, i.a. Per Martin-Löf, so I have great confidence in their suggestions. Also, Here is a review of the book.)
Another book I found useful, that was among the suggested reference literature for the logic course I took was
4.) Thomas Jech, Set Theory, Third edition. Springer 2000
I think both 3 and 4 above are rather standard, but the others may be less well known, and are actually the ones I prefer (however 1 not being purely set theory). You should be able to read the TOC of each of these books through the links I provided above.
It's always hard to know precisely what someone is after when asking these type of questions, but hopefully this will be of some help.
(Just if you're (or anyone is) interested, but perhaps not that relevant: other suggested, perhaps less well known (also seemingly far more challenging), reference literature for the set theory course was