You can get an idea of cycle routes from
www.bikeroutetoaster.com - go to the Course Creator tab, move the map to Edinburgh, then choose the Cycle maps (buttons in the top right of the map). Then you can see cycle routes that go through Edinburgh.
Be aware a lot of them will favour traffic-free and traffic-light routes over main roads, so they might take a six mile detour over a big hill to avoid half a mile of a fast road. Some of them will also use things like riverside towpaths and shared paths so an all-out road bike with skinny wheels might not be suitable for some sections.