The uniform motion of chiral magnetic skyrmions induced by a spin-transfer torque displays an intricate dependence on the skyrmions' topological charge and shape. We reveal surprising patterns in this dependence through simulations of the Landau-Lifshitz-Gilbert equation with Zhang-Li torque and explain them through a geometric analysis of Thiele's equation. Our results provide a universal geometrical understanding of the dependence of the skyrmion's velocity and Hall angle on the skyrmion's topological charge, shape, and orientation. The generality of our approach suggests the validity of our results for exchange-frustrated magnets, bubble materials, and other materials.