For pubs, I know lots of people like Hole in the Wall just up from the Minster on Petergate. I like the Mason's Arms which is just a five minute walk from the Castle Museum.
We used to like to go to Caesar's which is an Italian on Goodramgate, but we don't live that close to it any more. We are also big fans of Thida Thai on Rougier Street. My friend from Greece really likes the Turkish place also on Rougier Street. We've also eaten at Bari's which is in the Shambles and that was pretty good. If you don't mind chains, go to Ask for the atmosphere because it's the old Assembly rooms (think Jane Austen books with dances).
For lunches, I like the Early Gray Tea rooms for a high tea, much more food and I like it better than Betty's also in the Shambles. There's also a coffee shop called Cafe Concerto which is across from Hole in the Wall and is a bit expensive, but really has a fun atmosphere.
We don't eat out tons and usually we don't often go to eat in the pub any more.
Plus, I think more and mor of the pubs in York are now owned by the big chains or at least many seem to have a similar looking menu and you can often go to other pubs in the UK that have the same menues.