I want to start studying quantum mechanics, and then move to quantum field theory. I have a strong mathematical background, and I think this aspect of quantum physics won't be a problem to me. Though, Please note that I'm not a physicist and I have taken just one short course in university about classical physics, and I don't know anything about it. I ran over the Internet, and I saw that a lot of people debate over this. Some say that you don't need classical physics to understand quantum physics, others says you need to learn it in order to better understand quantum physics. Therefore, I thought that this is the perfect place to ask it. I would like to learn your opinion about this. And if you think that classical physics is a must or very useful for quantum physics, please specify me a single book, possible under 400 pages, which covers all the materials that I will need from classical physics.
This very much depends on what you want to do in the area of quantum theory. If you want to solve specific mathematical problems and to have only a very rough conception of why you are doing what you are doing, then you can in principle omit classical mechanics.
But if you want to have a well-rounded knowledge of the subject, you should know some basics of theoretical mechanics. The single book I recommend for a mathematician is
V. I. Arnold: Mathematical Methods of Classical Mechanics
Quite surprisingly, the textbook is available online. The most important concepts you should learn about are Lagrangian mechanics, the stationary action principle, Hamiltonian mechanics and their symplectic structure. Every single one of these concepts is crucial in quantum theory.
To understand why the symplectic structure of classical mechanics is so important you can read e.g. the canonical book:
P. A. M. Dirac: Principles of Quantum Mechanics
To understand the importance of the stationary action principle see any book on the path integral formulation. The general importance of Hamiltonian mechanics will become imminent already in quantum mechanics, the Lagrangian is very important throughout quantum field theory - you will encounter this in any textbook on the topics.
Both ways are possible. Since you seem to be a mathematician, let me try an analogy from mathematics. Say that you are accomplished in commutative algebra. Now, you want to study algebraic geometry. Sure, you can start with sheaves of local rings and cohomology of schemes, instead of "at the bottom" with classical algebraic varieties defined by polynomial equations, but that approach to start "from the top" will make it very difficult to actually understand the "geometry" part. Also, you might find it very hard to get an intuition about how to think about certain things if your intuition from commutative algebra fails.
The same is true in quantum mechanics. Studying analytical mechanics, where the motivation is always from classical intuition gives you an idea for the meaning of Hamiltonians, Lagrangians, wave functions and the sort, which gives you a certain intuition of how and why quantum mechanics evolved. You also learn why some simplifications may be OK, while others shouldn't be. Most importantly, learning classical mechanics and then quantum mechanics gives you some idea of where your classical intuition is works in quantum mechanics and where it doesn't.
Also, since a lot of the concepts simplify, learning the lingo of physics in this easier setting, might make you feel more comfortable with the ideas. Essentially, quantum mechanics is just a noncommutative version of classical mechanics and it's important to know how this comes about.
If however, you are mostly interested in axiomatic quantum field theory and don't care about physical meaning, then by all means, don't go to study classical mechanics, because you can't learn too much about the mathematics of QFT from it and you may well start by accepting the QFT axioms and go from there.
As for books, have a look here: Book about classical mechanics