Here is the full list of all 20 Beastrun achievements.
reach the fields
reach the forest
reach the meadows
level a skill up to level III
level a skill up to level IV
level a skill up to level V
level a unit up to level 2
level a unit up to level 5
level a unit up to level 10
get a super rare unit
clear the final room in a dungeon
level a relic up to level 5
find the smelly hat
Heal 100 health in one battle
Deal 40 damage or more in a single hit
execute 30 skills in the first 5s of battle
win a battle with 25% of enemy skills failed
deal 50 combined fire damage in one battle
win a battle that lasted at least 90s
defeat an opponent with combined unit levels that are at least 10 higher than yours