I went there a few years ago - it was very interesting.
Bars: The Grand, Pivni, Evil Eye, House of Trembling Madness, The Hop
Shops: Shambles - very quaint & Whipmawhopmagate :-D
Walks: City walls, Ghost walk, Museum Gardens, River walk to Millennium Bridge
Food: Star in the City, Blue Bicycle, El Goucho, Betty's (if you like that kinda thing)
Museums: Jorvik Centre, National Railway Museum (free), Art Gallery, Chocolate Experience
Cycling stuff: Your Bike Shed, Bic y mas (both fantastic bike cafes)
River stuff: Hire a boat, big river cruises, enjoy a pint by the river (Kings Head, Plonkers, The Lowther, Vodka Revs)
Cinema: City Screen (they have an outside terrace bar too)
Little bit further out of town is Bishopthorpe Road - some fantastic cafes and a great bike shop
Apart from the above best thing to do is get there early and have a good mooch around all the shops and go down all the side streets. Don't forget Clifton Moor and Monks Cross for shopping, also the Designer Outlet which is pretty decent - got Paul Smith, Hugo Boss etc rather than just full of GAP etc. Oh, for the best fish and chips go on the A64 towards Scarborough and visit Thompsons!