Although it's impossible to see all of Edinburgh in 3 days, you can get a fairly good feel for the city during visits of less than a half-week.
Day 1: The Royal Mile & Old Town
After checking into a hotel or guest house, head straight to the Royal Mile in Edinburgh's Old Town. It stretches from Edinburgh Castle to the Palace of Holyroodhouse. Along the way you can stop at the historic Gladstone's Land or St. Giles Cathedral, as well as the modern Scottish Parliament building. Take your time and be sure to wander down some of the narrow alleys and passageways that extend down the hill on either side of the Royal Mile, like ribs from a spine. Also, if you have time left over, check out the Grassmarket and take a detour to the Museum of Scotland.
Day 2: New Town & Scotland's National Galleries
Having a grip on Edinburgh's Old Town, it's time to move to the city's New Town, which dates to the late 1700s. Begin with the Princes Street Gardens and then move on to George Street, with its panoply of shops and stylish bars. Climb up to Calton Hill at the eastern end of New Town for the views and from the western side of the district, take in Charlotte Square or wander down to Stockbridge on the Water of Leith. Depending upon your legs, you can follow a path along the Water of Leith to the Dean Gallery, part of the capital National Galleries of Scotland. Catch the shuttle bus to any of the others, including the National Portrait Gallery or the main National Gallery of Scotland back in Princes Street Gardens.
Day 3: To Leith & the Southside
It is your last day, so let's move out of the city center and head to the sea. First stop, however, is the marvelous Royal Botanic Garden, where you might well spend a few hours wandering about the verdant paths. In Leith, you'll find the original port of Edinburgh, once an independent town in its own right. Diehard fans of golf should see Leith Links, one of the sport's historic landmarks or, if you are Royal fancier, go to see the Britannia, the yacht that Queen Elizabeth and family used until 1997. Come back towards central Edinburgh, but detour south to the Meadows and see some of the fine residential neighborhoods of Marchmont or Bruntsfield and get an idea of how Edinburghers live.