EVERY vs. ALL in set theory

I have had a tough time understanding the reasoning of axiomatic set theory. I suspect that one of the reasons is that in my mind ALL is equated with EVERY, and that for infinite elements that is inappropriate.

IN particular, I have a tough time figuring how the Axiom of choice is independent of the axiom of subsets, sum Set, and of replacement.

Anybody can point me to an author / resource that takes the time to go over the delicacies of working with infinite arguments?