Use the Hop-on/Hop-off buses to get around the city. Best value is the combination 48 hour pass. For about 38£ each you can use buses on any of the four lines and in addition get entrance tickets for several of the most popular Edinburgh sightseeing attractions along the route - Edinburgh Castle, Holyroodhouse Palace and a tour of the Royal Yacht Britannia.
Not only do you save a little money, but you avoid additional lines at the attractions, which can grow long later in the day, depending on the time of year. Tickets must be be purchased in advance from the booth on the Waverly Bridge and are valid for 48 hours from the time purchased (so technically if you buy them around lunch on a Friday, you could use them until lunch on a Sunday).