What I’m interested in trying to understand is how roots of polynomials distribute themselves in the affine space we are working over. I’d like to understand why the roots of certain polynomials form fractal shapes, or more generally why they distribute themselves in certain geometric ways.
My question is what mathematics do I require to be able to understand the geometric properties of roots?
I’m currently studying metric spaces, real analysis and ring theory in college and am wondering what else I would need?
I think that is enough to get started. You may not need to wait for more course work. Begin reading articles on the topic. When you see mathematics you don't know, look up the particulars, perhaps ask focused questions here.