If you are into Brutalist architecture (and I'm told some people are) this is the place to get your fix of blocky, grey, concrete buildings. Ziggurats are good though and there is plenty of green space. A walk around the lake gives a nice view of the university campus. People come to go fishing, walk their dogs or feed the ducks. Not a tourist highlight (unless you have a concrete fetish) but worth checking out if you are staying in the area for a while longer. Combines well with a trip to the Sanisbury Centre for Visual Arts which is located on the campus.
Calls itself a public art museum. Has exhibitions and a permenant collection. Not an essential place to see if you are on a short trip (unless you are very into art) but an interesting place to spend a few hours if you have longer to explore the region. I would recommend combining this with a look around the University of East Anglia campus. The Sainsbury Centre is actually on the univeristy campus. It is open to all though!
The Norfolk and Suffolk Broads is a unique area of water, grazing marshes, fen and woodland, and home to some of the rarest plants and creatures in the UK. It is Britain's largest protected wetland, having similar status to a national park.
To get there is a short trip from Norwich. And when you are there you can hire a boat and cruise on the broads.
St. George's Tombland Church is quite close to Norwich Cathedral. It has an impressive interior, and is well worth visiting if you like ancient churches.
Seven generations before me, my Ives ancestors attended this church (when they lived on Tombland), and so I always like to visit the church when I'm in Norwich. As of 2003 services were still held in the church on Sundays, so see if you can attend if you get the chance.
Not far from Tombland and the Catherdral, at Colgate Street in Norwich you can find St. Clements Church. Eight generations before me, my Ives ancestors were active in this church, and some of their remains lie in the church. Services aren't regularly held here anymore (at least not in 2003 when I last visited), but the door is open for you to come in for some prayer or meditation in the quiet surroundings. The Ives and Harvey families, who lived in wealth and splendor on Colgate Street back in the mid 18th Century, put up some monuments to their deceased relatives in this church.