There is probably a closed question covering all this, if someone has the courage to go and dig it up.
Anyway, to study QM you need knowledge of differential calculus, matrices (and in particular Hermitian matrices), a good understanding of classical mechanics (Hamiltonian formulation, and the concept of "turning points").
To go further, consider Lie groups (and Lie algebras to go even further), operator theory, relativistic mechanics (i.e. special relativity), path integrals and a solid knowledge of complex analysis to go through all the mathematical methods used in QM.
Also note that there are two ways to approach QM, the physical one and the mathematical one. Of course, the mathematical one is much more rigorous, and is probably harder to understand if you start with it. The physical approach will slowly introduce the concepts, and explain why we need a theory of quanta (the downside to this is that mathematical aspects will often appear to come out of the blue - but this actually helps focusing on the physics).
My advice: pick up a book on QM aimed at undergrads (Griffiths, Gasiorowicz) and try to work your way through it. When you want to get really serious, pick a mathematical/theoretical physics book (Weyl, Kuhn, Teschl)