Princess Quay is basically like any other shopping mall in any other big city. All the names you'll find here absolutely anywhere else, just rearranged in a different order. Of course go here if you want the usual comfortable high-street names, but if you want smaller local names try Saville St, Carr Lane, Paragon Street or King Edward Street that all fan out from the main entrance of the shopping centre. The old town down the end of Whitefriargate has a few older quaint-ish shops hanging on in there and a good indoor market worth a look round. Check any local map board around town or pick up a Hull Tourist Guide - they detail the streets I've mentioned.
One interesting thing to note about PQ though is the fact that it's built on stilts raising it up out of the former Princes Docks. It's architectuarally interesting too - the angular design is supposed to resemble the sails/hull of a ship. Blink, or come back after a few drinks and you might agree more.
Check out http://www.princes-quay.co.uk/ for the Mall's justification.
It's a toll bridge if you're in a car, but on foot or by bike it's free! You get a good view up and down the river from on the bridge, but don't try crosiing the road while on the bridge. Very dangerous, and not allowed!
There's a country park right below the Humber Bridge, where you can enjoy a bit of the coutryside whilst being beneath a motorway! The riverside along this area (Hessle Foresore) is pleasant too, with a good pub and restaurant.