For restaurants I would head to Wapping Wharf. There's a good selection of independent eateries catering for most tastes and they are all housed in shipping containers, with outside seating as well, so it's really quirky and very Bristol. I wouldn't personally bother with the Ivy, Browns or the Giggling Squid as they are chains and Bristol is famous for independently owned places. I do however second Pasture and Urban Tandoor but check to see if you can get a table with Pasture! Muse in Prince Street is also excellent - French/Indian fusion but it works. Also Mugshot is good for steaks.
Definitely walk around the Harbourside and catch a ferry or do a boat trip. M Shed is a free musuem on the Harbourside and will give you a feel of the city and its history. The SS Great Britain is nearby and worth a visit.
Also definitely go to St Nicholas Market and the Corn Exchange. Lots of independent stalls and an amazing collection of food stalls covering different cultures. Matina is my favourite.
It's against the law not to go to Clifton and see the famous suspension bridge. Clifton village is all cobbled streets and mews houses and really pretty and the view from the bridge is fantastic. You'll find good bakeries and coffee shops there. Primrose cafe in Boyce's Avenue is good for a sunny day and to people watch. Also visit Clifton Observatory with its camera obscura and caves.
Visit Gloucester Road which has the most independent shops in the whole of the UK. There are some interesting cafes and restaurants there too.
It's also against the law not to do a Bansky Tour - he was from Bristol in case someone still doesn't know! There are also ghosts tours and foodie tours you can find on Google.
Wake the Tiger is an interesting experience.
If you want cocktails head to Milk Thistle or Hyde & Co or Mugshot restaurant has a cocktail bar called The Library.
There's also Georgian House and Red Lodge which are period properties and museums. They are well worth a visit and give a real flavour of the city's past.
Enjoy your visit!