Well, here's what it seems like: We had the naïve set theories of Cantor (unaxiomatized) and Frege (first axiomatization), and everything was right with the world. Then, however, it eventually became clear that there were paradoxes that made the theory inconsistent, so Zermelo and Russell (and others) both set to work on coming up with alternative axiomatizations to deal with the paradoxes. Russell's Theory of Unramified Types ended up being unusable (cf. the proof that 1+1=2 in Principia Mathematica (the full proof is something like 100 pages). However, Zermelo was able to recover a good deal of naïve set theory in the Z (ZC), or Zermelo (resp. Zermelo + Choice) axiomatization (Zermelo+Choice).

However, it eventually became clear that Z(C) was too weak in some regards, so Fraenkel and Skolem both proposed (independently) the axiom schema of replacement to improve the power of the theory. If you include the axiom schema of regularity (which says that $\in$ is well-founded), you get ZF (resp. ZFC).

Most mathematicians accept as given the ZFC (or at least ZF) axioms for sets. These are treated at an intuitive level. Using set theory they define languages, axioms, theories, models and all the logic toolbox. Then they define (formalized) set theory again, using this language.

ZFC is built on first-order logic. Mathematicians then can construct *models* of languages and set theories internal to ZFC. (To answer your comment: The reason why we want to axiomatize over first order logic is because first order logic has all of the properties one would want from it, completeness, compactness, etc. In particular, the reason why we must first define theorems and proofs is because we need a proof calculus to be able to formally derive results.)

The second point of view is typical of logicians. They realize that in order to talk of logic they don't need the full power of set theory, so they take logic as God-given instead. Then set theory is formalized in this framework.

In fact, this is true for both mathematicians and logicians (although this view is strangely more popular among mathematicians than logicians). This would be the "bootstrapping" step for mathematicians to get to ZFC. Even though Bourbaki's book on set theory has an inadequate and confusing approach to set theory, the idea in the first few sections is correct. One first defines a what it means for something to be a theorem, what it means for something to be a proof, etc, then gives the axioms of formal logic, then constructs a set theory above that (this was one of the areas in which Bourbaki was the most influential as well. That is, his opinion on the logical grounding of set theory (but then not actually worrying about it once it's developed) is perhaps one of his most important lasting contributions to the world of mathematics.