Nice! Been there several times, going back in June actually.
I would recommend the Castle Museum, the Yorkshire Museum, York Minster, the National Railway Museam, Yorks Sweet Story, a visit to the shops in the Shambles, a walk along the roman walls. There's also the tour bus that goes around the city and the tour boats on the river, but I'd only recommend those if the weather is good.
If your looking for a good drink there's a couple of good cocktail bars to look out for, Bora Bora and Evil Eye. Plenty of good places for eating too, depending on what your into. The Blue Bicycle, Rustique and The Olive Tree are all good. Harkers is a decent pub, which also does food. Red Chilli is a good Chinese. Gourmet Burger Kitchen does a very good selection of burgers. Betty's Tea Room is a bit on the expensive side, but I think its worth it for the experience along. Their traditional Afternoon Tea is quite filling.