A lot if $G$ is transitive. Then $V=G/H$ for a subgroup $H$ (if it has a point), or a $G$-torsor mod $H$ (if it doesn't). Then most questions about the geometry of the variety are best answered by studying the group action. For instance, we can study line bundles on a flag variety of a reductive group using the root lattice for that group.

Another case where you gain a lot of information is where $G$ acts almost transitively, i.e., there is a dense orbit, as in the case of toric varieties. Then it is not as simple to "obtain" our variety as just choosing a group $G$ and subgroup $H$. We must also include some information on how to glue on the other orbits. But usually, because of the extra symmetry the group structure provides, this description is not so complex as defining an entire algebraic variety, since if you know something about the geometry of a point, you can deduce the equivalent statement about all other points in its orbit.