Wander round the harbourside and pop into the new M Shed museum, and then onto the SS Great Britain (a bit expensive to get in but you can use the tickets for a year).
Also go and see the graffiti on Nelson Street -See here for details – a quite amazing sight. Nearby is the Corn Street / St Nicholas Markets area with some nice old buildings, and the markets are worth a visit and have some great food stalls in them.
As above the Suspension Bridge is a must, and you could have a wander round Clifton Village which has some quite twee shops and a several nice cafes / restaurants for lunch (try the Primrose Cafe on Boyce’s Avenue)
A wander up to Cabot Tower on Brandon Hill (off Park Street) will provide some fabulous views of the city.