What are the equations governing stellar evolution (Luminosity, Mass, Temperature, Radius) I'm looking for ('simplified') equations governing stellar evolution. Especially how mass, luminosity, temperature and radius of a star change during it's lifetime. As well as equations which tell you how long a star stays in a certain stellar stage (PMS, MS, ...).

Extra clarification by example:

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After that a star with a mass between x and y enters the main sequence for a duration of $\Delta T = formula$. During this phase it's mass changes according to $M = formula(t)$. after that it enters the xxx  phase

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For those who think this question is to broad, please give some suggestions, 'google terms', ... from which I can move forward.
What you want is hydrodynamic stellar evolution. The following starts with a brief review of hydrostatic case to give the fundamentals. The last paragraph discusses the hydrodynamic case.

For hydrostatic case, see this, or this. Simplified stellar evolution assumes hydrostatic equilibrium (i.e., time-independent fashion). The system of equations compose of i) mass conservation, ii) mass transportation (i.e., hydrostatic equilibrium), iii) energy conservation, and iv) energy transportation. The system also requires i) the equation of state, ii) opacity, and iii) nuclear energy generation for being solved in a close form. 
The close-form solution is typically in the form of polytrophs (i.e., Lane-Emden equation). More complicated model, which is still hydrostatic equilibrium, will include other factors such as composition, rotation, and binary interaction. Since it is static and equilibrium case, the time-dependent terms are zero. 

For hydrodynamic case (i.e., time dependent) as you wanted, there is no close-form solution, therefore only hydrodynamic simulation is the method for the study. The simulation model uses the set of equations mentioned above without dropping the time-dependent terms. See, e.g., MESA.