Here is the full list of all 32 Pedigree Tactics achievements.
Log 10 species in your bestiary
Log 50 species in your bestiary
Log all species in your bestiary
Win one battle
Win ten battles
Win one hundred battles
Fuse one monster
Fuse ten monsters
Fuse one hundred monsters
Fuse a monster that can deal damage of every element
Fuse a monster that knows both Encourage and Discourage
Earn one million total experience
Recruit your first monster
Recruit ten monsters
Recruit one hundred monsters
Deal one hundred damage with a single attack
Deal one thousand damage with a single attack
Deal one million damage total
You have achieved everything there is to be achieved.