Here is the full list of all 10 Monster Harvest achievements.
Defeat Rigley.
Defeat the queen.
Unlock the fishing rod.
Unlock the furniture shop.
Apply blue slime to a crop.
Repair the bridge to the rec center.
Apply green slime to a crop.
Recieve a Planimal from the Professor.
Apply red slime to a crop.
Begin your first day on the farm.