I don't understand how everything relates. It seems that ZFC is a "first order theory" with axioms described in the language of first order logic, and it can recreate all the same axioms of Peano arithmetic (but not the other way around), so I suppose this makes PA a first order theory as well.
But then I am hearing that Peano's axioms are technically a second order theory? But then there's the first order theory that isn't as strong? Then I am unsure where natural numbers are defined exactly, and if this technically requires us to have set theory first in order to talk about membership? And what about functions? Don't these require set theory as well? Does this mean functions require ZFC? And if not, then what exactly are the "sets" we're using here?
I'm just totally lost as to what's defined where in terms of what and what's required to do this or that, it's all so hazy and vague and unclear and after reading countless answers on this website where everyone recommends the same unclear links that only partially answer the question, I'm losing a bit of hope.
Can anyone just plop the stuff down in a super easy to understand relationship hierarchy that clearly delineates what builds on what?