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