Several generations of mathematicians, working and otherwise, learned their portion of axiomatic set theory and foundations from the appendix to John L. Kelley's General Topology. It was still in print the last time I checked, and I remember liking it a lot.

Note on provenance. "The system of axioms adopted is a variant of systems of Skolem and of A.P. Morse and owes much to the Hilbert–Bernays–von Neumann system as formulated by Gödel."

If you like that sort of thing …