Here is the full list of all 14 Luna's Fishing Garden achievements. It takes around 2-3 hours to unlock all of the achievements on Windows.
Jump off the island into the ocean 3 times
Clean up the archipelago
Catch every fish
Catch an oarfish
Plant 15 lotuses
Build 10 bird nests
Complete any of Nigel's field journals
Feed Sam avocados
Help Gustav rebuild a bridge
Find Poppy in a cave
Use balloons 3 times
Catch the same fish 3 times in a row
Fail to catch 10 fish
Catch every fish in one spot