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