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