Suppose a compact Lie group G acts on a compact manifold Q in a not necessarily free manner. Is there any general method to gain information about the quotient Q/G (a stratified space)? For example, I would be interested in its (co)homology groups. To be a little more concrete - I am interested in G=SU(2) acting by diagonal conjugation on the space Q=G^N (N a positive integer).
Computing the cohomology of the quotient by a non-free action $X/G$ is generally quite difficult. As a first step, one can switch to a problem that is somewhat easier, namely that of computing the cohomology of the homotopy quotient, $X \times_G EG$. Here $EG$ is a contractible space with a free action of $G$. The idea here is that $X \times EG$ is $G$-equivariantly homotopy equivalent to $X$ and now the action is free. When $X$ is a point then the homotopy quotient is just $EG/G = BG$.
The cohomology of $X \times_G EG$ is what is known as the (Borel) equivariant cohomology of $X$, written $H_G^*(X)$. This is now a nice cohomology theory satisfying appropriate axioms. An important feature of it is that it is a module over the ring $H_G^*(pt) = H^*(BG)$.
If the group $G$ is a torus then there are some quite powerful tools for computing equivariant cohomology in terms of things like the set of nonfree orbits. A beautiful place to start is the paper of Aityah and Bott, The moment map and equivariant cohomology. Topology 23 (1984), no. 1, 1-28.
One of the theorems you will find explained in there is that the inclusion of the subspace of non-free orbits $S$ into $X$ becomes an isomorphism on equivariant cohomology once you localize (in the ring-theoretic sense) by inverting the equivariant Euler classes of the normal bundles of the components of $S$. Fortunately, the map from equivariant cohomology to the localized equivariant cohomology is usually injective so doing this doesn't lose any information.
Finally, there is a map from $X \times_G EG \to X/G$ given by collapsing $EG$ down to a point. This map gives rise to a spectral sequence that allows one to compute the cohomology of the quotient from the equivariant cohomology and additional geometric data about the action and its fixed points and stabilizers.
In general, this is a difficult question. Here are a couple of related facts that I know. Consider a discrete group $G$ acting on a space $X$, which we will assume is a simplicial complex (and that the action is simplicial). Moreover, to simplify things assume that the stabilizer of a simplex stabilizes that simplex pointwise (this can be arranged by subdividing).
1) If $X$ is simply-connected, then $X/G$ is simply connected if and only if $G$ is generated by elements that stabilize vertices. More generally, let $H$ be the subgroup of $G$ generated by vertex stabilizers (observe that this is normal!). There is then an exact sequence
$$1 \to H \to G \to \pi_1(X/G) \to 1.$$
This is a theorem of M.A. Armstrong; see his paper
MR0187244 (32 #4697) Armstrong, M. A. On the fundamental group of an orbit space. Proc. Cambridge Philos. Soc. 61 1965 639--646.
A related theorem can be found in my paper "Obtaining presentations from group actions without making choices".
2) As far as homology goes, there is a whole theory of equivariant homology here. A good first place to read about it is Brown's book "Cohomology of Groups", Chapter VII, and a more comprehensive introduction is tom Dieck's book "Transformation Groups"
As you will see if you read the above sources, the answer comes down to "It's complicated!". In concrete settings, you are probably better off trying to get a good topological/geometric understanding of the orbit space with your "bare hands".
Your example ($G^N/G, G=SU(2)$) is the space of conjugacy classes of $SU(2)$ representations of the free group of rank $N$. I seem to recall there are some articles out there that calculate its cohomology for small values of $N$, so you might look for those (of course, when $N=1$, you get a 1-simplex).
Eg. "The topology of moduli spaces of free group representations" http://www.ams.org/mathscinet-getitem?mr=2529483
But I think I recall some explicit results for $G=SU(2), N=2,3$ somewhere. Since the free group is a punctured surface group, you might look at articles about moduli space of flat connections on surfaces, and perhaps the Narasimhan-Sheshadri-type results which relate $U(n)$ and $SU(n)$ flat moduli spaces to moduli of holomorphic bundles over Riemann surfaces. I dont know if the punctures can be dealt with in that context.
You might look also at Akbulut-McCarthy's book, and I'm sure Andy P. and Jeffrey G. can direct you to other articles in this direction.