Classical mechanics is the study of the motion of bodies based upon Isaac Newton's famous laws of mechanics. There are no new physical concepts in classical mechanics that are not already extant in other areas of physics. What classical mechanics does is mathematically reformulate Newtonian physics to address a huge range of problems ranging from molecular dynamics to the motion of celestial bodies.
As one of the oldest branches of physics, it has long ago been displaced in many fields of study by newer theories (the foremost of these being quantum mechanics and relativity), but classical mechanics is far from being obsolete. Classical mechanics is very useful for analyzing problems in which quantum and relativistic effects are negligible, and its principles and mathematics are the foundation upon which numerous branches of modern physics are founded (including quantum mechanics and relativity). And finally, it is fascinating field of study unto itself—or at least some people think so. Maybe you'll be a fan of classical mechanics too, after having studied it.
The reader should be comfortable with Newton's laws and with basic physics concepts such as mass, moments of inertia, length, force and time (q.v. basic concepts). In addition, math is the crucial tool of physics, familiarity with geometry, algebra, and calculus is a must. In particular, the reader should be comfortable with multivariable calculus (if you do not know the difference between '∂f/∂x' and 'df/dx', then it's time to spend some quality time with a math textbook).
That said, mathematics is tool for physics, and only a tool. As much as it is important for the study of physics, physics is more than a mere exercise in math. It is also about finding different ways to look at the physical world, and developing intuition about how to predict natural phenomena ("how things work"). Readers need not have understood everything that was ever taught to them in a math course.
Suggested books to study
There is an extraordinarily large number of textbooks in theoretical mechanics, because it is a fairly old and well-studied subject. You need any textbook on classical mechanics that you can understand and that talks about "Lagrangians" early on. (Books that only talk about accelerations, forces, and torques may be quite advanced but they do not cover the subject of theoretical mechanics.)
- H. Goldstein. Classical mechanics. - Has everything standard in it and quite a few advanced topics. An old classic.
- L.N. Hand, J.D. Finch. Analytical mechanics (Cambridge, 1998). - A fresher, more didactic exposition of mechanics. Standard material.
- L. Landau, E. Lifshitz. Mechanics. -- A short, clean, concise treatment of mechanics.
- V.I. Arnold. Mathematical methods of classical mechanics. -- Has almost nothing standard in it but is excellent for a more mathematically minded student.
Part 1: Core material
- A brief review of Newtonian physics
- (Introduction and overview)
- A brief primer on differential equations
- The Lagrangian formulation of mechanics
- Exercises in setting up Lagrangian functions
- Constrained systems and Lagrange multipliers
- Overview of applications and further directions
- Motion in a central force field
- More on Lagrangian formalism
- Small oscillations
- Symmetries and conservation laws
- Mechanics of rigid bodies
- Non-Inertial Reference Frames
- Hamiltonian formalism
- Special Relativity