Here is the full list of all 39 Core Keeper achievements.
Hatched your first pet from an egg
Play the Tune of Tempest
Fed cattle
Reached max level on a pet
Gained 100 skill points in cooking
Gained 100 skill points in mining
Gained 100 skill points in melee
Obtained Soul Seeker
Obtained Phantom Spark
Gained 100 skill points in running
Gained 100 skill points in crafting
Obtained Rune Song
Gained 100 skill points in gardening
Gained 100 skill points in vitality
Defeated Ra-Akar the Sand Titan
Planted 10 Cherry Trees
Defeated Morpha the Aquatic Mass
Defeated Igneous the Molten Mass
Caught a Starlight Nautilus
Sat on a Caveling Toilet while wearing a crown
Gained over 200% movement speed
Talked to a Big Larva
Defeated Omoroth the Sea Titan
Discovered 100 recipes in the cook book
Harvested a golden plant
Defeated Ivy the Poisonous Mass
Defeated Malugaz the Corrupted
Defeated Ghorm the Devourer
Defeated Azeos the Sky Titan
Defeated The Hive Mother
Defeated King Slime
Defeated Glurch the Abominous Mass
Equipped the Ring of Rock and Ring of Stone
Gained 100 skill points in fishing
Gained 100 skill points in range
Obtain the Oracle Card deck
Stored all color variations of one pet type in a chest