2nd the Betty's tea room!
Bridge street has a slug and lettuce on the east side of the bridge and 'the Parish' (the church that's now a bar) is just a short walk over the same said bridge.
Coney street has a pitcher and piano which on the river and next to that revolution bar (its okay but can be a bit rowdy on weekends). Short walk away is all bar one which is very nice.
In terms of slightly quirkier bars / pubs on stonegate there is Delirium Tremens which is a pub above a shop in an old medieval building they do a range of belgian beers which are all very tasty. Evil Eye Lounge is also on stonegate and used to be quite a good chillout bar however i think it may have recently changed or is no longer there.
The 'micklegate run' (starting at gibson's (a rock bar) and heading down micklegate to bridge street) is quite popular but the bars aren't great and pretty much fit in the Yates kind of bar category.
A few more people might chime in with there 2p worth.