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