Advice – How do I go about self-study?

I apologize if these types of questions are not well received here. I thought that would be a better place than Quora, and I do not have many friends doing advanced math. I thought I would ask here …

I'm in my third year of school and I want to learn math. I want to do my doctorate, but from now on it's high school. I've had positive experiences with math that motivated me to keep going.

(1) When I was in the 9th grade two years ago, I received an arithmetic book for Christmas. I opened it in the middle of science and started to learn limits. I could fully understand boundaries, derivatives, and integrals. During that time, I was only in Algebra I and we had not even begun factoring squares. I quickly studied exponential functions and logarithmic functions and had a small introduction to trigonometry.

(2) In the 10th grade I dealt with trigonometry and was able to solve separable and linear order 1 ODEs. I could also understand concepts that resembled topological spaces (though I did not have many problems … I know, waste of time). Yes, I skipped most of Calc II, but that was a choice.

(3) I was able to keep up with Baby Rudin (even though I did not get far), as well as with texts in abstract algebra, metric spaces, etc.

In the 9th grade I took algebra I, last year geometry and algebra II, and this year I skipped Precalculus and take AP Calculus AB. I have a B (thanks to small, stupid mistakes) and everything has been checked with little to no difficulty so far. I want to get into advanced mathematics as much as possible and be competent in everything the university has to offer, but I get contradictory advice. I like math, which focuses on arithmetic, not very much Really I do not want to count alone when I do not need it.

I think I should make it clear that I am doing it Not I am familiar with single and multi-variable calculations and have not dealt in detail with differential equations or linear algebra. I am very interested in advanced mathematics, and the things that currently bother me most are homologous algebra, general topology, real analysis (I and II), and lying groups and lying algebras.

What do you suggest?