GO FOR IT, NEWBIE!!!

On grass, lava, and underground, there are no penalties. But here, as for all factions, for each terrain type, units have their own bonus of +1 to speed on native terrain.

This bonus only applies in combat. It has no effect on the hero's movement on the map.
I.e., scout heroes will run at the same speed on grass, lava, and underground — whether on a centaur, a wolf, or a harpy