I think you're looking for the Hochschild-Serre spectral sequence.  It's slightly more complicated than a long exact sequence, but you can extract the very concrete "inflation-restriction sequence" out of it.  

Your general question is a little too general to get a good answer, though there are some good other questions on this site that you will probably find very helpful, e.g.,

https://mathoverflow.net/questions/10879/intuition-for-group-cohomology

https://mathoverflow.net/questions/12539/essential-theorems-in-group-cohomology

If you really only want $n=0$ and $n=1$, these are very concrete and addressed in any of the standard references for group cohomology.  $H^0$ is the fixed-point functor, and $H^1$ is the group of "crossed homomorphisms" (which reduce to regular homomorphisms when the action is trivial).