Think there was an Edinburgh restaurant thread not so long back with enough recommendations for a years eating out. It really is a great city for restaurants – La Garrigue is a fine French one I've been to recently, 31 Jeffrey St, just to get the ball rolling.
It's also a great city for bevvying. I would eschew the tourism in favour of a skinful of ale at the various pubs in the centre – a fantastic way to see the city. If you like decent ale then The Cloisters by the Meadows, down from Tollcross, is a nice spot.
If you absolutely insist on going to see stuff, I would call in on Roslin chapel on the way up from Yorkshire – round the bypass a bit then down a couple of miles to the village of Roslin. I'm not usually excited by walking round historic buildings, but this one really is superb IMO.