When your Pet Peave becomes an adult, you will be able to feed it constantly with free food to increase its stats. Do this often to get a monster with as high of a rating as possible.
Growth
Focus on getting your Pet Peave from child to adult state as soon as possible. In each state, your monster will have a specific need (for example, playing with rubber balls, eating various things, or listening to music). Meet those needs to boosts the stats of your monster. When you are ready to grow your Pet Peave, shake your device. Even after growing to adulthood, monsters like to play and listen to music. It will not appear as a need above their head, but you can still try an activity. Each time your monsters enjoys an activity, it will receive a stat boost. Once your Pet Peave monster reaches adulthood, send it to battle other monsters.
Items
New items will appear in the store regularly. Purchase all of them, as they are required to increase the stats of your monster. Do not purchase multiple items of the same type (excluding food), as their bonuses will not stack.