at.algebraic topology - de Rham Cohomology of surfaces

For an orientable 2-dimensional surface, it's best to view it as a Riemann surface. Any standard reference on Riemann surfaces (in my day, this was Gunning's Lectures on Riemann surfaces) will work out the deRham cohomology, as well as its decomposition into Dolbeault cohomology. And then if you look in, say, Griffiths and Harris, you can see how to compute the deRham cohomology of complex projective spaces in any dimension.

For standard manifolds that are quotients of compact Lie groups, I believe you can compute deRham cohomology using averaging. I don't know where to find this, though. I vaguely recall learning this from notes by Bott.