After some time spent on it I now have some understanding of de Rham cohomology and can actually calculate some cohomology groups. However I have now and then encountered many other cohomology theories: compact, harmonic, rational, Cech ...

I understand that they all share a similar structure, however it seems to me that the calculation techniques can be quite different: for example homotopy invariance is a key tool to use in de Rham cohomology, but it is not valid in compact cohomology.

So, is it that at a higher level of knowledge the similarities between the various theories are such that some experiences in computing cohomology groups in one of them can be easily transferred to the others or is just life hard?

I should point out that my aim is certainly not to perform research in this area, but rather to be able to calculate the cohomology groups of some non-trivial space (but nothing terribly complicated) when the need arises.