Classical mechanics is sometimes considered a branch of applied mathematics. It consists of kinematics, the description of motion, and dynamics, the study of the action of forces in producing either motion or static equilibrium, these topics are important to know: introduction to mechanics, including Newton's laws, work, kinetic energy, potential energy, the conservation of energy, momentum, collisions, rotation and rotational motion, gravitation, and periodic motion. In addition to variational principle and Lagrange's equations, central force problem, rigid body, oscillations, and Hamilton equation of motion.

Here I recommend studying: Mechanics by Keith R. Symon (Undergraduate), Classical Dynamics of Particles and Systems by Stephen T. Thornton, Jerry B. Marion (Undergraduate), and Classical Mechanics by Herbert Goldstein, Charles Poole, John Safko (Graduate). Also you can see Prof. David Tong free lectures of Dynamics and Relativity  and Classical Dynamics. 

By this point, you should have finished the introductory calculus books and are ready to move on to more advanced mathematics. You should start working through linear algebra, complex analysis, real analysis, partial differential equations, and ordinary differential equations (See Page of Mathematical Methods in Physics)