Could you please explain how does a general pulley system work and how I can solve problems associated with those?

Pulley systems allow for force transmission from one body to another or more. Intelligent pulley systems can be used to apply more force to a body of interest than that generated (by any power system or other means).

In general, solving a problem where a pulley system is involved, entails the calculation of the acceleration of the bodies of the system and the tension of the cable(s) that holds them together.

To do so, it is first necessary to draw the free body diagram of EACH body (i.e. a simple representation of the external forces acting on them). This usually involves the weight of each given body, frictional forces, and the tension force from the cable.

By means of Newton's second law, the final acceleration of each body can be expressed in terms of the external forces.
Thus, you will (usually and hopefully) end up with as many equations (given by the number of bodies that constitute the system) as unknowns, which you can therefore easily solve for through simultaneous equations.