Here is the full list of all 38 Lighthouse Keeper achievements.
Unlock the Light House and enter the first time
Get axe
Add the whale amber generator to get endless energy
Get Pickaxe
Get Fishing Rod
Get Speargun
Get Vacuum Gun
Get Container
Get Navigator
Get Sling
Go to Pirate's Haunt for the first time
Go to Volcano Island for the first time
Go to Frozen Island for the first time
Go to Withered Island for the first time
Go to Azek Meteor Island for the first time
Go to Ascension Island for the first time
Go to Flytrap Island for the first time
Go to Archaic Mine Island for the first time
Go to UpperSea Tavern for the first time
Go to Fruitfulness Island for the first time
Go to Rib Island for the first time
Get your first harvest
Cook for the first time
Use recycle machine for the first time
Read a book for the first time
Make your first potion
Catch fish five times
Kill a shark
Upgrade a tool for the first time
Repair a tool for the first time
Trade for the first time
Use mill for the first time
Get Re Moraja's health under %25
Find the room behind the waterfall
Open light of Lighthouse for the first time
Get all the stabilizers
Find your first burial
Get the ship