TITLE: Toy sheaf cohomology computation
QUESTION [21 upvotes]: I asked this question a while back on MO : 
One thing that really helped in learning the Serre SS was doing particular computations (like $H^*(CP^{\infty})$)
I am curious, as a sort of followup if anyone can suggest:

A reference where small computations are carried out? or
A specific computation to do with a small enough sheaf an some simple topological space that would be able to give one a feel for sheaf cohomology. So this space that we are working over need not be a scheme, in fact it would probably be best if it were not a scheme since I don't understand them quite yet. And are there tricks of the trade to computing these things? or do people just hammer away ate injective resolutions? 

In short, please suggest a space and a sheaf on it that I should work on computing the sheaf cohomology of.
PS: I of course welcome any other suggestions for understanding how to compute sheaf cohomology.

REPLY [3 votes]: This is rather scheme-y, but there's  a really nice paper by Kempf (hopefully you have institutional access :() that gives a very basic and elementary proof that the higher cohomology of a quasi-coherent sheaf on an affine scheme is trivial. The first part of the paper uses nothing more than the basic properties  (e.g. long exact sequence) of cohomology, and might be fun. I thought it was fun, anyway; it's also nice because it shows that Hartshorne is unnecessarily restrictive in sticking to noetherian affine schemes in chapter III (even if one wants to avoid anything fancy).
OK, update: here is the proof explained (admittedly by a beginner :)).