It's hard to say, where to start, because you didn't tell us, what you already know. On the physics side you should be pretty familiar with classical (Newtonian) mechanics in its formulation in Hamiltonian canonical formalism and Poisson brackets. On the math side you should have a good knowledge of linear algebra, including complex vector spaces with a scalar product and some calculus, including Fourier transformation.
Then I'd recommend to start with modern books that do not use the historical way of teaching QT. A good example is J. J. Sakurai, Modern Quantum Mechanics, which our professor of the QM 1 theory lecture recommended. I think that's a pretty nice starting point with the right balance between math and physics. You should not get involved too much in the formalities of functional analysis in the beginning, but later it is good to have at least some knowledge about the subtleties of, e.g., operators with a continuous spectrum, the rigged Hilbert space, etc.