Anglesey Abbey.
Use the park and rides, there's 5 sites all around the city. (Unless you're staying centrally?)
All the parks are nice, midsummer common is right by the river. Nice to watch the towers go by etc.
Lots of interesting little side streets to get diverted by.
Heffers Bookshop is great.
Make a resolution to not go to any chain cafes/restaurant.... there's hundreds of options outside of Costa/yo sushi etc. Honestly, just wander around the corner (the town centre is about higgeldy piggedly) and there'll be a tiny little café or bakery or restaurant or market stall selling good food.
Check the theatres. As PP said, the open air Shakespeare is a real treat and is on during the summer.
Jesus Green lido was 24° yesterday so not absolutely freezing.
Re shopping centres, the Grafton Centre is more high street brands, Grand Arcade/Lion Yard is a bit more high end. (On the whole)
Massive Library in town centre, if you like that sort of thing.
Fitzwilliam museum is brilliant.