Royal Mile
Walk down the Royal Mile from Edinburgh Castle to the Palace of Holyroodhouse. The Royal Mile is the backbone of the sixteenth century Old Town] and most of the buildings date from this period. Take the time to divert from the main route down some of the many small "closes" that run off the Royal Mile to each side. There are plenty of bars along the length of the Royal Mile, where you can relax over a pint of Edinburgh Ale and/or a classic single malt.
Holyrood Park
Walk around Holyrood Park which is just East of the Old Town. Created by King James the VI in the 16th century, it is like a Scottish landscape in miniature, containing Arthur's Seat (http://www.scottishsport....), an extinct volcano with wonderful views from the summit, three lochs, and the stunning Salisbury Crags.
Meadows
Visit the Meadows, Edinburgh's modest version of Central Park. Always awash with activity, the Meadows is the perfect place to relax, read a book, get drunk, or play some football. Tennis courts are available for hire at the eastern end of the park.
Follow a ghost walk around the back streets surrounding The Royal Mile and learn about the sinister goings on of Edinburgh's past. These tours are generally excellent and are highly recommended.
There are free walking tours that set off periodically from the High Street Starbucks. They are in a variety of languages and take about 3 hours, with a stop for lunch. Tickets are free, but the tour guides appreciate tips. Even with a generous tip for the guide, these are still a far cheaper way to see the city than most bus tours, and they give you a much more personal experience.