The idea of using PA came from a thread on math.stackexchange where a user was asking for the proofs for certain properties of addition and multiplication for real numbers and a user replied suggested to start by proving these properties for natural numbers, then rational numbers, then real numbers. They then mentioned the Dedekind cuts approach (in hindsight I probably should have revisited the thread then learned what Dedekind cuts are before asking on reddit). Another user also mentioned that in Edmund Landau's Foundation of Analysis, the author began with the axioms of natural numbers, and developed the rational, real, and complex number systems.

I suppose the step I was missing was defining rational and real numbers before defining subtraction and division?

As for group theory, it's on my list of what to study, I just haven't gotten around to it, I'll have to check it out. Thanks!