Visit Cabot Tower near the top of Park St, which also has lots of trendy shops. You can't miss it because its the steepest road in Bristol city centre!
Visit the Clifton suspension bridge, and if you can find the entrance, the little tunnel that leads to a balcony half way up the cliff face. (very near the suspension bridge.
The Downs and Ashton Court park are nice for chilling out.
Depending on how shop happy you are, you could got to the new Cabots Circus in the centre (consumer chain stores etc w/horrible green/purple lighting...)
Take a wonder up Gloucester Rd. there you will find nuff shops with records, graffiti and other 'cool shit'. Lots of cafes etc.
There is the SS Great Britain and John Cabots sailing boat replica on the far side of the harbour. You will pass the Industrial Museum on the way, and at the far end, there are some nice pubs to eat at with views of the harbour.
For art, visit the Arnolfini. Across the footbridge is also Watershed cinema and the @ Bristol centre. @Bristol is a very hands on science museum with a botanicle garden, giode cinema and IMax cinema.
Club wise, The Blue Mountain does dubstep nights. Other clubs playing non cheese include Lakota, Trinity Hall (occassionally), Easton Community Centre (occassionally), The Black Swan pub. There are others, but I haven't lived there for a while, so you'll need to check flyers/listings prior to arrival.
For more hip hoppy stuff, jazz and folk, check out the Thekla boat or Cosies bar. For jazz and folk, head to the The Llandogger Trow and 'the Duke' (directly across from the Llandogger). For rock music, the Hatchet pub, the Fleece & Firkin, the Corft or the Louisiana should do the trick...
After that, its more down to timing. The pirate caves near the harbour open once a year to the public. Bristol has a strong maritime history, and traded in tabacco and slaves. Bristol is a town built on top of a town, so many hidden passage ways and structures exist under modern Bristol. For instance, the river Avon runs directly under the city centre for a few miles reapearing by the IKEA in Eastville.
Otherwise, check out Venue magazine for local club listings, and the tourist information website for places to visit!
plus American Adventure was fucking shit anyways, get back to ya chickens 
