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