If you want to level up either your character or your monsters, the best way to do it is to farm EXP. You can farm EXP through a number of methods, but the best by far would be to farm quests that give a decent amount of XP and bosses/monsters when exploring areas.
You can accept quests via the quest board that unlocks early on in the game, and you can also repeat pretty much all of them. Another cool thing about quests is that they will also display on the bottom right of your screen how much EXP and zenny they give, thus making it easier to find one to farm for EXP.
We highly recommend choosing the ones that give the most or ones that involve fighting monsters and bosses, as you can earn additional experience from defeating both.
That being said, that's only half of what you can do to farm EXP and level your monsters and your character quickly. As we stated before, the next method would be to explore and fight the various monsters and bosses you can fight out in the wild.
Doing so will be a good source of experience and materials, which will prove useful for making both armor and weapons in the game.
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