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, 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.
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.
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.)
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
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, reference literature for the set theory course was
Also, perhaps not customary practice, but I must add that: looking through the suggestions by (and by) Asaf Karagila above, these look really good; thank you for those, I will also surely read them.)