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