Here is the full list of all 39 Core Keeper achievements.
Planted 10 Cherry Trees
Defeated Omoroth the Sea Titan
Talked to a Big Larva
Play the Tune of Tempest
Defeated Ivy the Poisonous Mass
Sat on a Caveling Toilet while wearing a crown
Hatched your first pet from an egg
Fed cattle
Harvested a golden plant
Defeated Malugaz the Corrupted
Defeated Azeos the Sky Titan
Defeated King Slime
Defeated The Hive Mother
Equipped the Ring of Rock and Ring of Stone
Defeated Ghorm the Devourer
Defeated Glurch the Abominous Mass
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
Caught a Starlight Nautilus
Gained over 200% movement speed
Discovered 100 recipes in the cook book
Stored all color variations of one pet type in a chest
Reached max level on a pet