Unfortunately, my school only teaches classical physics
Good to hear that, because you cannot understand the physics of quantum theory without a very solid understanding of classical physics. That understanding will have to go well beyond the level of classical physics taught in high school.
There are certainly mathematical tools that you need as a minimum in quantum theory, but these tools don't help you if you do not understand the core physics really well.
An firm understanding of the concepts (not the mathematics) of Thermodynamics, Electromagnetic theory and Classical mechanics will serve you a great deal more than knowing the fancy mathematics. Why ? Because the reason you end up with equations of a particular form in quantum theory is because of the need for the results to conform to these other concepts.
You also tend to pick up the mathematics you need for quantum theory in learning these things.
If you want to make a "small investment" in mathematics (that won't interfere too much, if at all with your high school studies) I would recommend learning :
- Basic of Group Theory
- Basics of Complex Numbers
- Basic calculus - differentiation, integration
These are all typically taught to a useful level in higher level High School mathematics courses these days.
So what you might do is (maybe) talk to your High School mathematics teachers and ask if you could get a start on the higher level mathematics I mentioned.
G Smith mentioned that it's extremely important you put equal (if not more) effort into the non-physics and mathematics because you need the good results in those (at the very least) to get into university or a technical school. You also need them because, in the real world, having these other skills turns out to be rather important in terms of getting good jobs, being able to perform well at business (even physicists have to do interviews, business meetings, write reports, and so on). They round out your skill set and help a lot - don't neglect them. To put this in some perspective, my university physics course had mandatory courses in business skills (e.g. basic accountancy) which also came with a requirement for mandatory passes and contributed to your final degree grade. These things are important and the skills served me well.