I'm very far from knowing much about this. Evidently people who have looked at Euclid have realized that his axioms are far more murky than realized. Hilbert has an axiomatization of Euclidean geometry that's regarded as better. This is a direction for you to go in.

https://math.stackexchange.com/questions/328028/what-are-the-differences-between-hilberts-axioms-and-euclids-axioms
Likewise, the ancient parallel postulate has now been replaced by

Playfair's axiom.

Like I say, I don't know anything about any of this; these are just some things I've run across that will be of interest to you.

I definitely commend you for reading Euclid.

All of modern math has been axiomatized by the Zermelo-Fraenkel axioms of set theory, called ZF. Sometimes it's called ZFC in honor of Mrs. Fraenkel, who was pro-choice. That's a lame joke.

https://en.wikipedia.org/wiki/Zermelo–Fraenkel_set_theory
All of the object of modern algebra are axiomatized. Look up groups, rings, fields, monoids, semigroups, vector spaces, and modules.

There are axioms for the real numbers. There are the Peano axioms for the natural numbers.

There's an axiomatic description for pretty much everything these days.