I've just finished High School and I've always been very interested in math.
Yesterday I was proving some very basic theorems on Geometry (Pythagorean/Hero's, sine/cosine laws/). At some point I started to pay attention to the steps I was taking and realized that I had assumed much knowledge, many other theorems to complete the demonstrations. They just seemed to be intuitively true to me, like the proportionality theorem, or assuming a square had 360° and so forth.
I know that a lot of what I was concerned with was rigorously demonstrated by Euclid in his Elements, but I want something more modern that uses algebra and arithmetic for the demonstrations.
I'm also afraid that my knowledge of Algebra, Arithmetic, Set Theory, and anything involving mathematics is unfounded. Where should I start in order to make my knowledge rigorous and well-founded? Maybe I should learn some mathematical language?