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There are more complicated things like cones. (Open sets in) Weyl chambers might be present, for example. One thing that I know, is that the orbit space of a finite groups action is stratified into submanifolds which correspond to orbit types corresponding to cojugacy classes of isotropy groups inside the finite group. There is always one big connected and locally connected open dense stratum, the regular stratum. I can point you to sections 29 and 30 of here. Maybe, also this paper is of interest to you (there are some mistakes in it). And this other paper is devoted to Riemannian geometry of orbit spaces of isometric group actions in general.