Edinburgh is a Great place to visit of course, but there are some nice attractions outside the city such as the Craigmillar castle. Further to the north the Islands are very recommended to visit, and the coastal cliffs are also a magnificent landscape to see. Just choose an Island and go there. In Orkeny Islands you can find stone pillars very similar to the stone henge on the mainland.
I'd be happy to give some more tips if you just specify what are you interested in.