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