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
Milk a Cow 100 times
Catch one of every fish
Cook one of each Preserve
Bake one of each Cake
Bake one of each Pie
Catch one of every bug
Catch 100 Fish
Gather 100 Mushrooms
Cook one of each Soup
Mine 1000 Raw Stone
Find 100 Mixed Seeds
Catch 100 Bugs
Reap a Gold Seed
Deliver your farm registration letter to the mayor
Gather 100 Shells
Throw a ball
Chop 1000 Raw Wood
Harvest Star Jelly
Use a wardrobe to change your outfit!
Sweep a ball
Loot a forest treasure chest
Harvest Grapes, Artichokes, Purple Nettle and Rhubarb
Cook Couscous and Baked Artichoke with Goats Cheese
Catch one of each fish found on the Galaxy Island
Catch one of each bug found on the Galaxy Island.
Collect Goats Milk and Brown Pheasant Feathers