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