This can be completed on the first island or as you progress naturally to other islands. Enemies are essentially anything that isn't wildlife or NPC. The first enemy you encounter will be a blue slime with a white circle on it's face called a mono, defeat 100 of them or a combination of them and the skeletons in the starter dungeons.
Around the island mono can be found.
Dungeon of the Island: Skeletons can be fought.
Dungeon of the Island: Lizards can be fought.