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