Hey Pudding -- there are lots of side quests to complete if you use the explore button everywhere. I don't have a list, but I know you can find several quests (that aren't species-specific) in Gateway City, Homely Hamlet, Pixpet City, Bumbly Beach, Barrow Woods, Jungle of Sweetness, Golden Dew Mountains, and Endless Sea. The only place that doesn't have side quests, if I remember right, is Navady Island. Most of these quests will get you evolution items for vivid pets, but some raise your social level (which helps you access Bumbly Beach). Some may be ability-specific, but many aren't.

Let me know if anything I said was confusing or if you have more questions!