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.
Shear a Sheep 100 times
Milk a Cow 100 times
Catch one of every bug
Catch one of every fish
Gather 100 Mushrooms
Catch 100 Fish
Plant 1000 Seeds
Bake one of each Pie
Bake one of each Cake
Sweep a ball
Cook one of each Preserve
Mine 1000 Raw Stone
Cook one of each Soup
Reap a Gold Seed
Gather 100 Shells
Chop 1000 Raw Wood
Catch 100 Bugs
Find 100 Mixed Seeds
Loot a forest treasure chest
Harvest Star Jelly
Throw a ball
Use a wardrobe to change your outfit!
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