For a simple complex group G and Riemann surface X, are the (integral, if possible) cohomology groups of the moduli of holomorphic G-bundles on X written down somewhere, either explicitly or implicitly? If not, are there some specific cases where these groups have been calculated?
See Atiyah-Bott, "Yang-Mills Equations over Riemann Surfaces" for the case of stable holomorphic vector bundles. See in particular section 9 and Theorem 9.11.
They compute the Betti numbers of $N(n,k)$, the moduli space of stable holomorphic vector bundles of rank $n$ and first Chern class $k$, when $n$ and $k$ are relatively prime. They also find generators for the cohomology ring. These "Atiyah-Bott generators" are the Künneth components of the Chern classes of the universal bundle. In this paper they don't compute the relations for these generators.
The main tool in this paper is Morse theory.
I think the reason for restricting to stable bundles is to get a moduli space (rather than some kind of stack or whatever). The reason for fixing the first Chern class is just because different Chern classes correspond to different connected components. I think the reason for the condition that $n$ and $k$ are relatively prime is just to ensure that the moduli space is a smooth manifold.
Since these moduli spaces are also algebraic varieties, one can also compute their Betti numbers using the machinery of the Weil conjectures... There are some comments about this, as well as references, in the introduction of the paper and in section 11.
EDIT: I just did a Google search and found this paper of Heinloth and Schmitt, which claims to compute the cohomology ring of the entire moduli stack: http://staff.science.uva.nl/~heinloth/HS_v4.pdf
This paper says that the Atiyah-Bott generators are in fact free generators...