Here is the full list of all 38 Potion Permit achievements.
Forage 1,000 stone
Unlock all achievements
Forage 1,000 tree
Work part-time in the Post Office 15 times
Successfully cure 50 patients
Collect 10 type of fish in Moonbury
Cook food 15 times
Cook food 50 times
Work part-time in the Church 15 times
Collect 10 pieces of furniture
Clear 75 quests from the Community Board
Failed to cure 10 patients
Pet your dog 50 times
Date an NPC for the first time
Clear 25 quests from the Community Board
Forage 1,000 plants
Reached Friendship Lv.3 with a villager
Work part-time in the Police Department 15 times
Failed to cure patient for the first time
Roll for 1,000 times
Defeat 1,000 enemies
Reach Friendship Lv.2 with 10 villagers
Collect 10,000 stone
Brew potions 150 times
Have a bath in Willow Waters Bathhouse 10 times
Collect 3,000 wood
Cook food 5 times
Go fishing 50 times
Successfully cure 10 patients
Collect 10,000 coins
Brew potions 50 times
Reach Friendship Lv.1 with 5 villagers
Reached Friendship Lv.1 with your dog
Passed out for the first time
Use fast travel 20 times
Brew potions 15 times
Brew your first potion
Start a new game