Since you're a high-school student, here's an answer that's less sophisticated and much less rigorous:

I suppose you could make up any set of axioms you want, and start using them to prove theorems. So, as you say, you could make Pythagoras' theorem an axiom in your world, and then you wouldn't need to "prove" it.

But, if you're going to start making up your own system of axioms, and doing mathematics in this private world, there are a few things you need to worry about:

(1) If no-one else uses the same axioms as you, then no-one will be very interested in your "theorems", since they are only true in your private world. Your private world might be a bit lonely. So, better to use the same axioms as everyone else.

(2) It's useful (though not absolutely necessary) to have a system of axioms that bears some relationship to reality. That way, the theorems you prove will sometimes give you information that has value in the "real" world -- in fields like economics and engineering, for example. Your private world might be quite different from physical reality, if you don't choose the axioms carefully. So, your results could be misleading or even dangerous, even though they are provably "true" in your world.

(3) If you're not careful, the system of axioms you invent might lead to contradictions, or it might have other fundamental logical flaws. The axioms can't be completely arbitrary (as far as I know).

There are some areas of mathematics where part of the game is making up modified systems of axioms and seeing what happens. But most of us play by a fairly well established set of rules, for the reasons outlined above (and for other reasons, too, I expect).

**Additions**

Regarding your added comment that "there is a false backbone of rigour that seems true until you start questioning the very fundamentals". It seems to me that the rigour is in the reasoning that's used to derive theorems from the chosen set of axioms. I don't think this rigour is "false".

What's bothering you, I suppose, is that there is some freedom when choosing the set of axioms, and, depending on what choices you make, you get a different set of theorems -- a different version of the truth, and different statements of what is "right" and "wrong". I understand your concern -- I can see how it might be disturbing to find out that the axioms of mathematics are not universally agreed. One example of a debatable axiom is the "Axiom of Choice" (read more here). Most mathematicians assume that this axiom is true, but some don't, and, of course, the two groups get a different set of theorems. Not entirely different, but different.

But, on the other hand, the choice of axioms is not completely arbitrary, and there is a very large overlap in the sets of axioms that are in common use. So, in practice, things typically work just fine, despite the fact that the foundations are not entirely cast in stone.

Questioning the fundamentals, as you are doing, is a valid thing to do, and mathematicians have been doing it for a long time. If you want to know more about this, from sources that are at least somewhat "credible and reliable", then this Wikipedia page might be a good place to start.