Hello Max,
let me provide you with yet another opinion:
Imho the best way (and probably also the most fun way) of learning mathematics is not so much by learning mathematical theories from books (you will learn enough theory during your studies anyway) but by actually doing mathematics and by that I mean solving mathematical problems. A good oppertunity to do so is to compete in mathematical competitions (Check out this list from wikipedia http://en.wikipedia.org/wiki/List_of_mathematics_competitions). Not only will it provide you with challenging problems but also it is a good oppertunity in the later stages of the competitions to meet other guys who have the same passion about mathematics as you. It is a great feeling to know that one is not "the only one".
But you were asking for books so I give you my all time favourite:
László Lovász - Combinatorial Problems and Exercises
This books contains nothing but mathematical problems from rather easy ones you can figure out in a quarter of an hour to really challenging ones which could take you up to a week or more to solve. As background you only need basic linear algebra, probability and calculus. Solving all of these problems (of course without looking into the solution before) would probably teach you enough math to start tackling real unsolved problems in the beautiful field of discrete mathematics afterwards, i.e. to do real mathematical research. Sounds tempting doesn't it? ;)