If you want to truly understand QM at say, a graduate level (which I started with, after learning basic Newtonian mechanics and basic Lagrangian mechanics), then you could try to use Fitzpatrick's "Quantum Mechanics: A Graduate Course." Lots of it leaves exercises up to the reader (where it says stuff like 'it is obvious that...'), but it forces people to think, which really helps people understand the concepts. Now it IS a graduate level textbook, but it isn't that bad to move through at a decent rate (it's the summer now, and I've gone through about half of it, 80 pages, in three weeks or so). However, it really helps to understand linear algebra (what's a vector space? what are eigenvalues, etc.?). I disagree with JasonJo where you have to know linear algebra including Hilbert Spaces... it most definitely helps, but Fitzpatrick's book teaches you the Hilbert Space knowledge... it's more of something you just notice rather than pursue... (it helps to put it into perspective... a state ket is a vector which represents a quantum state... so Fitzpatrick just says okay * insert info about state kets here * oh and it's part of a Hilbert space * more stuff here *, etc.). In short, he just *mentions* that state kets are part of a Hilbert space. There's nothing really more.
Now, with the classical mechanics, there is a chapter relating to Hamiltonian mechanics, but it's not absolutely NECESSARY to read through Hamiltonian mechanics... just know the basics (the equations associated with it, and that's about it). Now, Newtonian mechanics is very important to know, because QM is a generalization of Newtonian + Lagrangian + Hamiltonian mechanics... so if you're not familiar with quantities like momentum, energy, and angular momentum, you may be in for a ride. Best of luck!
