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