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:
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
- Neilson & 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: Nuovo Cimento 4,224
If you want to convince youself 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.