If you have never been to Edinburgh before (and you sound as though you haven't) then a walk down the Royal mile is a wonderful way to spend half a day (or more).
There is a picture of it here
http://www.architecture.ed.ac.uk/assets/Ed1/ca stle_royalmile.jpg
It starts at Edinburgh castle in the foreground (in the picture) and then continues past some amazing buildings full of history, till you get to Hollyrood house that you can see in the distance (in the picture).
This has to be one of the most amazing and interestng stretches of road in the world.
Although you cannot see it in the picture, Arthur's seat is to the right of Hollyrood house, you can just see the start of Hollyrood park in the top right of the picture.
Web site for royal mile:
http://www.edinburgh-royalmile.com/