Using python to search a DF for names and return results from another column

As the title suggests, I would like to take a DF (appointment data) and get a function to run through the names and return the appointment types the person had in date order.

Image is attached of how the data would look... Appointment Data

The output would ideally look something like; [Type1, Type2, Type1] [Type2, type1, Type1]

Thanks