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