Here is the full list of all 39 Core Keeper achievements.
Equipped the Ring of Rock and Ring of Stone
Defeated The Hive Mother
Defeated Ghorm the Devourer
Defeated King Slime
Defeated Glurch the Abominous Mass
Defeated Azeos the Sky Titan
Defeated Malugaz the Corrupted
Defeated Ivy the Poisonous Mass
Defeated Omoroth the Sea Titan
Defeated Morpha the Aquatic Mass
Defeated Ra-Akar the Sand Titan
Defeated Igneous the Molten Mass
Gained 100 skill points in mining
Gained 100 skill points in cooking
Gained 100 skill points in fishing
Gained 100 skill points in gardening
Gained 100 skill points in melee
Gained 100 skill points in range
Gained 100 skill points in vitality
Gained 100 skill points in running
Gained 100 skill points in crafting
Obtain the Oracle Card deck
Obtained Rune Song
Obtained Phantom Spark
Obtained Soul Seeker
Talked to a Big Larva
Caught a Starlight Nautilus
Gained over 200% movement speed
Discovered 100 recipes in the cook book
Harvested a golden plant
Sat on a Caveling Toilet while wearing a crown
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