The requirement for leveling up in Save the World is earning experience. You do this by going out and doing missions, finding survivors to add to your group, and defeating enemies. The more that you do, the more experience that you will earn. Once you have a lot, take a look at your survivor heroes and you can select which one you want to level up there. At the same time, you have your general power level as well for your character. They level up as you gain experience and grants you skill points for the tree.