Here is the full list of all 22 Forever Home achievements.
Visited The Spot
Visit the continent of Tren
Visit the continent of Charitine
Visit the continent of Heyama
Complete the main story
Split a Shard by fully leveling it
Craft an item
Radiate one monster
Radiate all monsters
Complete Enda's sidequest
Complete Slash's sidequest
Complete Burns sidequest
Complete Asera's sidequest
Complete Sandra's sidequest
Complete Throg's sidequest
Complete Eidol's sidequest
Obtain the weapon Eternal Courage
Defeat Arc in the Stadium
Defeat Attrition in the Stadium
Defeat The Sickness in the Stadium
Defeat Tenjin
Defeat Final Tenjin in a New Game+