Although of course Kostya's answer is a good definitive one, I thought best to mention that there are tools available other than spectral sequences depending on exactly what you want.
For example, if $M$ is an orientable surface, $G$ is finite, the action is not neccessarly free and the quotient is still smooth, then you can use the Riemann-Hurwitz formula: http://en.wikipedia.org/wiki/Riemann%E2%80%93Hurwitz_formula.
There are also higher dimensional analogues of this which give you information about the canonical bundle of $M$ if $M$ is a complex manifold.
