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