I’m live in York so know a thing or two!
Have a walk around the Bar Walls as it’s a good way to see the city, don’t bother with the tour buses. If you want to go to Bettys go to the less well known one down Stonegate called Little Bettys, much shorter ques and as good. Cliffords Tower is ok but after 15 minutes you’ve seen it, like other people have said The Viking Centre is ok if it’s wet, it had a large refit a few years ago and supposed to be better (not been though so can’t judge) York Minster is an impressive building and well worth seeing. If you like trains The National Railway Museum is only 10 minutes walk from the city centre. The best advice is just wander about it’s a compact City. Look for a book called Snickleways of York. These are medieval alleys around the city worth exploring.
For a good restaurants try Fossgate there are quite a few nice ones down there, or go to The Go Down on Clifford Street as favourite of ours. Evil Eye is worth going to for drinks on an evening, huge cocktail menu for you and a good selection of beers for the missus 😉 if you’re into nightclubs go to Toyko on Toft Green, avoid all the others.