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