For first time visitors and those in Edinburgh for a short time, take one of the hop-on hop-off buses round the City.  It is an ideal way to see a good selection of Edinburgh - which is quite spread out and built on hills.  There are four different bus tours with varying routes.  Check to make sure the one you book goes to the sites you want to see, and/or passes your hotel.

The Royal Mile is the historic spine of the city, running from the Castle at its highest point to the Palace of Holyroodhouse at the bottom. In between there are lots of significant historic sites and lots of cafes and souvenir shops.   If you are short of time do not walk all the way down the Royal Mile.  Enjoy the Castle at one end, and hop on a bus to Holyrood at the other.     

Alternatively, keep in mind that  Edinburgh is a walker's city.  If you have only one day in Edinburgh, consider this itinerary or reverse it, depending on your hotel or if you want to go down rather than uphill. 

Get out of the door by 9:00 and go to Palace of Holyroodhouse (opens at 9:30).  If there is a good exhibit at the new and beautiful Queen's Gallery, buy a combination ticket and see the exhibit there first, then with your audio guide, tour the Palace and the Abbey ruins at your own pace.  All of this will appeal to travel-savvy children and teens.  Don't neglect a short stroll in the Palace gardens after your indoors tour (weather permitting). They are pretty and well kept. 

Then walk slowly uphill toward Edinburgh Castle stopping on the way to look at John Knox's house, and St. Giles Cathedral ); go inside these sights or not, as your whim takes you.  Peek into the Closes (small pedestrian alleyways) along the way, and if a shop strikes your fancy, of course, do that too.  There will be several shops offering free samples of W hiskey, try them if your age and inclinations permit.  Along the way have lunch, perhaps "toasties" or soup at a charming place like the Forsythe Tea Shop in one of the Closes.  A short detour up George IV Bridge will take you to the statue of Greyfriars Bobby and the nearby  Greyfriars Church

Be sure to listen for the One O’clock Gun which is sounded from the Castle ramparts.  Be sure to get to the Castle by 3:00 so you'll have enough time to see it all.  You can see the Honours of Scotland (crown jewels, etc.) and enough armour, dungeons and exhibits to entertain the whole family.  Sailors from the American Revolution were among the prisoners here, and you can still see their names and a ship with an American flag and other handiwork carved into the old wooden doors of the prison. The view over Edinburgh is marvellous, the complimentary guided tours or the audio tours are very informative. 

For the end of a perfect day and another fine view, walk up (you can take a taxi part of the way) to the top of Arthur's Seat for sunset.  You will revel in the beauty and stillness of the countryside around Scotland 's most beautiful city.