Where to go after Halmos' *Naive Set Theory* I'm in the process of finishing Halmos' Naive Set Theory, and I found the subject fascinating, so I would like to carry on reading about Set Theory when I'm done. 
From what I've been able to gather from similar questions asked in the past, the books which seem to have the greatest number of followers are A. Levy's Basic Set Theory and Introduction to Set Theory, by K. Hrbacek and T. Jech. Is any of these preferable over the other? I would be self-studying from them, and probably be asking on this site when I get stuck. 
Further, is the material presented in Halmos' book enough to tackle the books mentioned above? I should add that I have close to no background in logic. I'm comfortable with strings of symbols involving quantifiers $\forall$ and $\exists$, but I know nothing about, say, Godel's Incompleteness theorems. How much logic should I know to read any of the two books mentioned above? 
Thank you for your time!
 A: You will find a section on entry level set theory (§4.3), and then another section on more advanced stuff (§6.6), in my Teach Yourself Logic 2015: A Study Guide (an annotated list of some of the available books on different areas of logic). The most recent version is available here.
A: I'm also in the quest of learning this subject, and I have seen the following books.
There is a basic, beautiful book by Just & Weese suggested in this answer with some commentary on it. It has a brief intro into the logic you need to know to get involved.
A very nice introductory exposition is the book by Moschovakis; you can read thorough review here. Its historical comments are extremely interesting.
Finally, I think the doorway into serious set theory is the book Set Theory An Introduction To Independence Proofs (Studies in Logic and the Foundations of Mathematics) by Kunen. The 1980 edition is by now a classic, but the 2011 version of the book takes care of a lot of details that in the older book are left to the reader.
