Here is the full list of all 14 Grow Your Pets achievements.
Get your first Pet.
Feed your Pet.
Upgrade a Pet to the second stage.
Upgrade a Pet to the third stage.
Harvest food for the first time.
Catch a fish.
Find the Ritual.
Win a fight in the Forest.
Lose a fight in the Forest.
Find the Pet God.
Acquire an upgrade at the Trader.
Craft an item.
Spin the Roulette.
Complete a quest from the Notice Board.