Where to go after Halmos' *Naive Set Theory*

Solution 1:

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.

Solution 2:

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.