I suspect there may be a meta-theorem that says that (if not all, then at least most) "interesting or useful" partitions of a group $G$ arise in the following way. Take a group $H$ and let $H$ act on $G$ in some "interesting or useful" way. Then take the orbits of the action of $H$ on $G$ for your partition.
For example, the conjugacy classes of $G$ arise as the orbits of the action of the inner automorphism group of $G$ on $G$. This can be generalised by considering other subgroups of the automorphism group of $G$ (notably, the full automorphism group). If the group $G$ has additional structure (say, a group topology or an invariant measure), then one can consider just automorphisms that preserve that structure.
The cosets of a subgroup $H$ of $G$ arise as the orbits of the action of $H$ on $G$ by (left or right) multiplication.
The twisted conjugacy classes mentioned in the answer by @TastyRomeo arise from a twisted conjugacy action.
The partition mentioned by @Max in a comment comes from an action of a cyclic group of order $2$ acting by inversion on the group.