Leveling up only happens by using Stamina to get through the games 10 worlds, where you gain Experience to level. Eventually you can open up the Dark Dungeon (at level 40 I believe) that can also give experience. Along the way you'll get gear and stones dropped by enemies, which can also be leveled up but with Gold and Stones and Souls. Gear is more responsible for your atk/def going up, but leveling doesn't do nothing.