Which bus number from Shanghai to Taizhou (Jiangsu Province)

You can take subway line 3 at West Yan'an Road, get off at Main Railway Station, Exit 4. Then walk northwest around 600 yards (550 meters) to get to General Bus Station. There are several coaches that can take you to Taizhou. You can choose the schedules at 12:30 or 13:30.
As I know, no return bus at around 21:00. So you are suggested to stay at there for one night and go back in the next early morning.

Thank you very much for this very good answer. Is there a website where I can see the bus number and the timetable for such buses? I have changed my plans and now need to leave early mornings at 6-7 am...

So you shall leave from the university for Taizhou around 6-7am, right? Then high speed train can be the best choice. You can take a taxi to Hongqiao Rail Station with the fare of around CNY40. Then catch a fast train to Taizhou in 3-3.5h. The early trains are scheduled at 06:20, 06:40, 06:50, 07:00 and 07:10. Hope you can catch one.