The transition from classical to quantum is characterized by dropping the commutative law for the multiplication of physical quantities.
Quantum mechanics is the noncommutative version of classical Hamiltonian mechanics.
Quantum field theory is the noncommutative version of classical Lagrangian mechanics.
[Edit: This is just the big picture, explaining the most common stuff in a few lines. But the borderline between QM and QFT is fuzzy, and you can treat both QM and QFT with the main tools of the other field. For example, 1+0-dimensional QFT is equivalent to QM, essentially in the same way as classically, and QFT can be done via Hamiltonians (e.g., via similarity renormalization).]
Classical and Quantum Mechanics via Lie algebras
is all about the connections between classical and quantum mechanics
(but very little about QFT, sofar).
You'll find the big picture carefully motivated in Chapter 1, with the remainder of the book showing in increasing detail how it works. Most of the mathematics and physics you'll need to follow the book is introduced along the way, but knowledge of linear algebra and multivariate differentiation is assumed.