Don't learn classical set theory if you want to do category theory. There are a few "set theories" that are better-suited for category theory, for a list, check out the page "Set Theory" on nLab.
And also, you don't really need to learn set theory for category theory. While there can be size issues etc, arrow-theoretic language is completely different from set-theoretic language. The fact is, set theory is much easier and less abstract than category theory, and you don't want to settle into your nice set theoretic world just to have it all blown away once you start doing category theory.
For example, in set theory, all injective maps ($f:X\rightarrow Y$) (monomorphisms in the category of sets) admit a left inverse ($g:Y\rightarrow X$ such that $g \circ f = id_X$). This is not true in a general category. That is, the morphisms with this property in a general category aren't only monomorphisms, they're split monomorphisms, so set theoretic intuition absolutely fails here. There are countless other examples, but I'm sure you see my point.