You may begin breeding monsters with Clint in Herbage after defeating the ghost monster in his basement. Breeding uses two parent monsters, creating a new monster from them. You get three choices, one of each parent's type with random move combinations from the parents. The third monster choice is an entirely new monster, depending on which monsters were mated.
Breeding monsters is the best way to improve in the game, as they are near the parents' average levels, but with better stats and move combinations. It also allows you to gain new monsters you can't otherwise have, including the starter monsters.
Below is a list of monster combinations and resulting third offspring choice. (Still needs completion, please contribute when able.)
Sedisonic + Sedisonic yields Manpower (Animal)
Sedisonic + Spirispear yields Elevil (Rock)
Windicrest + Windicrest yields Kiticon (Fire Starter)
Windicrest + Sedisonic yields Fernarus (Plant)
Windicrest + Kiticon yields Bubblonster (Spirit)