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.
Clean up the archipelago
Complete any of Nigel's field journals
Catch every fish
Plant 15 lotuses
Build 10 bird nests
Catch an oarfish
Help Gustav rebuild a bridge
Jump off the island into the ocean 3 times
Find Poppy in a cave
Use balloons 3 times
Feed Sam avocados
Catch every fish in one spot
Fail to catch 10 fish
Catch the same fish 3 times in a row