Bakery - CS (Cosgrove & Son) - on Fishergate (between the Novotel and St George's Fields park), and also on Tower St, bang opposite Clifford's Tower. Everything is sourdough, and amazing.
Clifford's Tower for a bit of history and fantastic views.
Spark on Piccadilly for street food traders with a communal eating space
Panda Mami - world food buffet, just on the south side of the Ouse Bridge
The Wizard Walk of York is hilarious (gets very booked up), my 2 loved it at 10 and 13. If they are too cool for school, then The York Dungeon.
The Chocolate Story is interesting, with lots of freebies!
The Hole in Wand is indoor , wizard themed crazy golf. Good for rainy days.
A boat cruise is fun and a bit of a break from walking
There are also lots of people offering ghost walks and walks around the city walls. You can do the city walls easily without a guide, there are history boards as you go round. There are many entrances in the walls/gates etc. - just step in and away you go. There's also the Cat Trail - you can pick up a leaflet from Tourist Info.
The National Railway Museum is impressive even if you're not into trains/engineering, and mind blowing if you are. And free!
We had a great mini break to York last summer, and we only live an hour away.
The Blue Barbakan is a polish restaurant with an interesting history!
We also like The Double Dutch Pancake House
That's without the Minster, Jorvik, shops, museums, art gallery...