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