Here is the full list of all 20 Happy Quest achievements.
Complete chapter 15
Reach max level of Freeze skill
Reach max level of Shocking Ground skill
Reach max level of Eruption skill
Reach max level of Heal skill
Reach max level of MP Regeneration stat
Complete chapter 12
Reach max level of Power stat
Reach max level of Stamina stat
Earn 100000 gold
Complete chapter 9
Reach max level of MP stat
Complete chapter 6
Reach max level of HP stat
Purchase 1 skill slot
Purchase 2 skill slots
Earn 10000 gold
Complete chapter 3
Kill all monsters in 1 move!
Use Power Up system the first time