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