I've been to Edinburgh a couple of times now, I've done a lot of shopping in Princes Street, looked around the railway station, and spent an afternoon exploring the Castle that dominates the city.
A walk down the Royal Mile (which is longer than an "English" one) takes you to Holyroodhouse house, one of the Queens palaces in Scotland, and opposite is the Scottish Parliament building. There has been a lot of debate about this building - and I must admit, it's not one of my favourites. It seems to be a concrete block with the few sticks of wood stuck on the outside.
Just down from the Castle is the Camera Obscura, which allows you to see a moving image of the city displayed on a table directly in front of you. But that's not all, there's a great display of holograms, 3D images, and even a couple of remote control CCTV cameras that you can take control of - move them up down, and even zoom in - Great fun for the biggest of kids!!
Another thing you must try to do is visit the Royal Yacht Britannia - Okay, I know it sounds like a tourist trap, but it is very interesting to see how frugal the royal apartments are on board.