Does anyone know a good book where I can find the computation of the de Rham Cohomology of surfaces in R^3 and other classical manifolds (higher dimensional spheres and projective spaces for example) ? I found Tu's book "An Introduction Manifolds", where a computation is presented via Mayer-Vietoris sequences. However, it does not contain other examples. Does anyone know any other good material?
Is it important that it be de Rham cohomology? Because any algebraic topology book will compute the integral cohomology of all those examples and more (e.g. Hatcher's "Algebraic Topology", available free online); from this you can find the real cohomology by the universal coefficient theorem. For de Rham cohomology in particular, I would try Bott and Tu's "Differential Forms in Algebraic Topology".
For surfaces there's very nice and down-to-earth approach in Fulton's "Algebraic Topology" specially chapter 18 "Cohomology of Surfaces". For higher dimensional classical manifolds including the projective spaces see Karoubi's "Algebraic Topology via Differential Geometry" specially chapter V "Computing Cohomology". All of these use de Rham cohomology.
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.
I would like to recommend "From Calculus to Cohomology". Contains several examples and even a chapter 7 : "Applications of de Rham cohomology". It is also so-called "self-contained", but on the downside it does contain some minor flaws which can be quite confusing when reading the material for the first time. I think this is the book you are looking for.
A detailed exposition of de Rahm cohomology and it's uses is given here .
I don't remember exactly, but I think your question and in general your approach to topology should be along the lines of Bott-Tu book.
check these of J. Harrison and the ivancevics bros
these skip a little Mayer-Vietoris... (but believe me: it is unwise avoid Mayer-Vietoris :)