There are three major ways to earn money:
Use the Post Drop Box to sell potions.
Check the Town notice board each week and select requests from residents. You will have to gather specific items or ingredients in exchange for money or food items.
Complete three part-time jobs daily after receiving your first Approval Badge. These jobs are available in Service Buildings and are activated through cutscenes with specific characters. You can find more information about them here: https://potion-permit.fandom.com/wiki/Part-Time_Job