1) If you want to compute the rational or real cohomology of something, you can try to use rational homotopy theory. Rational homotopy theory says that the rational singular chain complex is (as a dga) chain equivalent to a very small chain complex, the minimal model, whose generators are in correspondence to the generators of the rational homotopy groups. So, if you have knowledge of the rational homotopy groups, you can try this. It works also quite well to study the cohomology of the (free) loop space of a space, because you can compute the minimal model of the (free) loop space of a space if you know the minimal model of the space.
2) If you can show that your space is a $BG$, its cohomology equals the group cohomology of $G$ which is computable in some cases. But I suppose that for manifolds the direct usage of this method is not very efficient since the fundamental group of all acyclic manifolds is infinite and group cohomologies of infinite groups can be very difficult to compute by algebraic means.
3) For a compact connected Lie group you can use the theorem that the De Rham cohomology equals the equivariant forms (see chapter V.12 in Bredon).
4) Even if you have no strict Lie group structure, but only a multiplication which fulfills the axioms up to homotopy, i. e. an H-space, you can make use of this structure. With field coefficients, you have the structure of a Hopf algebra on the cohomology/homology and there are various structure theorems. E.g. an easy application of this method is that an H-space which is a finite CW-complex with non-trivial homology has zero Euler-characteristic.
