I would really recommend going to The Botanical Gardens. The extensive gardens contain really tropical plants as well as typically English ones. It is a really nice place to wander around, and is not expensive. Situated on Trumpington Road (a little further up from the Fitzwilliam).
The Fitzwilliam Museum is an impressive building from the outside and has a range of paintings and a large collection of armour and Egyptian artefacts inside. It's free and close to the centre (go straight past Kings college in a straight line, you can't miss it).