"Descriptive set theory" is a good starting point, though it's the bulk of what set theorists in general do. It's true that there's an infinite possible set of

"Descriptive set theory" is a good starting point, though it's the bulk of what set theorists in general do. It's true that there's an infinite possible set of axioms. It does seem that the types of axioms that have consequences that humans are interested in fall into simple families. For example, many seemingly unrelated questions are settled by assume the existence of very large sets (larger than can normally constructed in set theory).