Go for a drink at the Guy Fawkes Inn, House of Trembling Madness also very good. On a nice day walk along the river - start at Skeldergate on the west side of the river, walk past Rowntrees park then over at millenium bridge, back up the other side to where you started at Skeldergate.
I agree with Skosh for dinner - do book well in advance. In fact that's my main York advice - book tables! Deans Court round the back of the Minster is very nice, they have a lovely garden overlooking the city wall to have a very civilised drink in. It's next door to Treasurers House mentioned above, that whole area behind the Minster is very nice and atmospheric, not too busy. They use it for filming period dramas quite a lot.
Micklegate (where Skosh is) is quite nice for a wander - lots of nice cafes now, including Partisan mentioned above.
If you're going to do Betty's, do afternoon tea, as that's the only way to book a table, then you can swan past everyone in the queue!
If you are going to the art gallery, I think they have a nice bar for the summer with a nice garden - not been yet but looks good.