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Non-relativistic Quantum Mechanics: Sakurai's Modern Quantum Mechanics
This is a textbook for graduate-level Quantum Mechanics. It's advantages over other texts, such as Messiah's Quantum Mechanics, Cohen-Tannoudji's Quantum Mechanics, and Greiner's Quantum Mechanics: An introduction is in it's use of experimental results. Sakurai weaves in these important experiments when they can be used to motivate the theoretical development. The beginning, using the Stern-Gerlach experiment to introduce the subject, is the best I have ever encountered.
I second the recommendation, although I haven't read other textbooks.
What are the prerequisites for reading this? What level of mathematics and background of classical physics?
You need some solid Linear Algebra: Vector Space, dual vector space, unitary and hermitian matrices, eigenvectors and eigenvalues, trace... Mind that you should learn these things with mathematical approach, for example, vectors are elements of vector space which has certain axioms, and not 3D arrows, like pupils learn in school. Since book has this approach (matrix mechanics, rather than wave mechanics), you don't need too strong analysis, you can just trust that some things are working that way, but if you want to understand it fully, i recommend taking some analysis course as well, to be able to understand decomposition in eigenfunctions. Integrals and derivatives are MUST, however.
Why don't you like Cohen-Tannoudji?
I found this book very good as well. I want to add a comment, though.
If you start reading it, and you get lost, just stop reading that chapter and go to the next one. Read this book lightly at first, then start clarifying everything afterwards. Reading introduction of every chapter first is very clever.