patom, I spent a weekend in York in March and LOVED it there. For roast beef and yorkshire pudding, I went to
Thomas' of York at 3 Museum Street. It was a Sunday afternoon (of course) and I was heading toward the National Railroad Museum and there it was. The ciders on tap were great too.
Others have mentioned
Betty's. So will I. You should go. Many go for high tea. I went for the Swiss menu at dinner time. They are also chocolatiers.
Lunch — I was walking across the Lendal bridge and I saw what looked like a little castle on the beginning of Museum Street. I forgot the name of this little café so I had to look it up. It's called
Hartley's Fine Sandwiches and I discovered it by chance walking from the train station into town. It's pretty much on the bridge on the York Minster side. Fabulous sandwiches—what they can do with bacon....
Stay at a B&B and enjoy their breakfasts!