Here is the full list of all 28 The Islander achievements.
Reach Level 20 in a Single Run
Buy a total of 250 Animals
Buy a total of 100 Animals
Plant a total of 250 Trees
Catch a total of 100 Fish
Plant a total of 250 Crops
Plant a total of 250 Bushes
Catch a total of 100 Starfish
Catch a total of 100 Parrots
Plant a total of 100 Trees
Have at least 1M Gold
Plant a total of 100 Bushes
Buy a total of 200 Cocobots
Buy a total of 25 Animals
Reach Level 10 in a Single Run
Have at least 100K Gold
Plant a total of 100 Crops
Buy a total of 100 Land Expansions
Build a total of 750 Decorations
Plant a total of 25 Trees
Plant a total of 25 Bushes
Build a total of 250 Decorations
Reach Level 5 in a Single Run
Plant a total of 25 Crops
Build a total of 100 Decorations
Cut down a total of 250 Trees
Break a total of 250 Rocks
Have at least 1000 Gold