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