Castle museum is great. You can easily spend half a day in there. I've been many times.
Viking centre is okay. I've never been that much of a fan and think it's a bit overpriced, but it's popular enough.
York dungeon my kids found a bit scary. The railway museum is fascinating if you're into trains. The boat trips and the hire boats are fun. The minster isn't too bad, especially going up to the top, though it's a bit pricey. Chocolate story is good. There are sweet shops and Harry Potter ships in the Shambles and a good open air for market behind the Shambles.
There are two gaming cafes as well. One in Gillygate and the other at the end of Coney Street, near the bridge, if they like playing board / card games. The Dig is also good as an activity based thing. Obviously there's the walls to walk around and there's an art gallery (bit dull) and museum and nice gardens
Personally I think the castle museum is the best thing for kids. They tend to have little competitions for kids and there is plenty they can interact with.
Food wise, depends what you like. There's plenty. I would recommend Licc if you like ice cream. It's delicious and there's loads to choose.
I know York very well, so PM me if you want any specific information.