Are you looking for suggestions for other things to do in Norwich, or for suggestions of other places to go in England during this 8 or 9 days?
I'd spend the bulk of that time in London, Norwich only needs a couple of days. Cambridge is the obvious stop between Norwich and London, can be done as a day trip, Depends if you like looking at old stuff I guess.
Americans tend to like to go and visit Oxford too, the other famous University city. That can be done as a day trip from London too.
Sheringham and Cromer are half an hour from Norwich if you want to visit a nice Norfolk seaside town and have some proper fish and chips. Depends on your typical pace of life and what you are in to I guess. Cheap trains to those places from Norwich also.
For your first visit to England I wouldn't get any grand ideas about trying to head up North, so much to see and do in London. You won't run out of things to see and do there, and most of the galleries are free entry too.
Try and squeeze in a non-league game somewhere too if the fixtures fit your itinerary, St. Albans is just north of London and Jamie Cureton is banging them in there - 7th tier of English football.
Depends what you are into and how you are getting about I suppose. Could easily get away without getting a hire car and just hitting the trains from London, but if you do hire a car at any point then there is a few national parks which suddenly enter into the equation.
I wouldn't hire a car personally, a nightmare driving and parking in London (where you don't need a car because of the underground system),and you don't need a car to get to Norwich or anywhere else due to our decent rail network.