Here is the full list of all 28 Staxel achievements.
The base game contains 23 achievements, and there is 1 DLC pack containing 5 achievements.
Catch one of every fish
Cook one of each Preserve
Cook one of each Soup
Bake one of each Cake
Bake one of each Pie
Shear a Sheep 100 times
Sweep a ball
Reap a Gold Seed
Catch one of every bug
Gather 100 Mushrooms
Milk a Cow 100 times
Use a wardrobe to change your outfit!
Mine 1000 Raw Stone
Plant 1000 Seeds
Catch 100 Fish
Chop 1000 Raw Wood
Gather 100 Shells
Catch 100 Bugs
Harvest Star Jelly
Find 100 Mixed Seeds
Loot a forest treasure chest
Throw a ball
Deliver your farm registration letter to the mayor
Catch one of each bug found on the Galaxy Island.
Catch one of each fish found on the Galaxy Island
Cook Couscous and Baked Artichoke with Goats Cheese
Harvest Grapes, Artichokes, Purple Nettle and Rhubarb
Collect Goats Milk and Brown Pheasant Feathers