I would suggest that you don't do any preliminary reading, and just learn QM directly. There is not much to it, the requisite background is very primitive linear algebra, and Dirac's book "The Principles of Quantum Mechanics" and Feynman's "Lectures on Physics Vol III" can be read with Wikipedia help without any prerequisites.

The classical mechanics you need to know is not very sophisticated either— you just need to know Newton's laws, and how they come from a Lagrangian or Hamiltonian, which is covered in standard sources. You don't need so much deep stuff, although knowing Poisson brackets is handy for seeing the vestigial quantumness in the classical mechanics structure.

I would suggest reading the following Wikipedia pages for a historical perspective, which helps a lot with historical literature:

• Old quantum theory
• Adiabatic invariant
• Correspondence principle
• Matrix mechanics

This is wrongly left out of most books, and this is a shame. There is no unified presentation of the historical material except on Wikipedia, and this is why these pages are up there. Once you get the historical stuff (it's not a lot), Dirac gives a conceptually self-contained introduction to the mathematics, the notation, and the physics, while Feynman is path-integral friendly, so you can go on to read Feynman and Hibbs, or Mandelstam and Yourgrau without any delay.

It is usually a waste of time to try to go through prerequisites, as these are usually boring and most of the material doesn't end up getting used. For QM, you need to come in knowing what a matrix is, and what an eigenvalue is, which is probably best learned from Dirac. Ron Maimon (https://physics.stackexchange.com/users/4864/ron-maimon), What's the standard "roadmap" to learning quantum physics?, URL (version: 2012-10-21): https://physics.stackexchange.com/q/41317

For quantum mechanics, the original is still the best:

• Dirac's "The Principles of Quantum Mechanics".

It's clear, it's terse, and it's comprehensive. All other books take most of their material from this source.

For a basic short introduction to quantum mechanics, you can't beat:

• Feynman Lectures on Physics Vol III

This is very good and intuitive, and complementary to the remaining books.

Landau and Lifschitz "Quantum Mechanics" is heavy on good exercizes and mathematical tools. L&L include topics not covered everywhere else. The standard undergraduate books on quantum mechanics are not very good in comparison to these, and should not be used.

A book which requires minimum of calculus or continuous mathematics is

• Nielsen & Chuang: "Quantum Computation and Quantum Information"

This focuses on modern research, and discrete systems in quantum computation. If you don't know calculus, learn it, but you might find this book the most accessible. It's long though.

On advanced quantum mechanics, there are good books are by Gottfried and by Sakurai. Berezin's book is also a great classic.

For the path integral, you can read Feynman and Hibbs, but I like Feynman's 1948 Reviews of Modern Physics article more. There is also a good book which covers the path integral:

Yourgrau & Mandelstam: Variational Principles in Classical and Quantum Physics. The original source for the Fermionic path integral is still the best, in my opinion:

• D.J. Candlin: Il Nuovo Cimento 4 no. 2, 231 (1956)

If you want to convince yourself quantum mechanics is necessary, you should recapitulate the historical development. For this, the following source is good:

Ter Haar's "The Old Quantum Theory" (it's short) to learn Bohr Sommerfeld quantization You can also read the Wikipedia page on old quantum theory for a sketchy summary, then look at the page on matrix mechanics. This explains the intuition Heisenberg had about matrix elements, something which is not in Dirac's book or anywhere else. Heisenberg's reasoning is also found to certain extent in the first chapters of this book:

Connes "Noncommutative geometry". This book is also very interesting for other reasons.Ron Maimon (https://physics.stackexchange.com/users/4864/ron-maimon), What is a good introductory book on quantum mechanics?, URL (version: 2015-10-02): https://physics.stackexchange.com/q/33260