To every city that I go to I do a free walking tour. It's absolutely priceless! You get to see so much in 2 hours, get to know the city, the main monuments and the history from locals. Just google "Edinburgh free walking tour" and you will have plenty to choose from. I recommend Sandeman's tours.
Just a quick disclaimer, they advertise as free but they survive on tips. So if you give them a few pounds at the end of the tour they'll be more than happy.