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).
- 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".
- 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".
