I've lived here my whole life being Scottish haha! I live in the NE now but I'm originally from Fife ( near Edinburgh )
We will be heading down to Edinburgh to see The Lion King and the traditional Christmas markets the second week in December for my sons Birthday, we always tend to stay in the Apex a short walk to the princess Street (city centre), the breakfast room has a beautiful view of the Castle but there lots of places to stay for all budgets, if you're feeling flush, The Balmoral or The Scotsman is they way to go but there are good deals in the lower budget hotels like the Holliday inn Express or Travel lodge. The jury's inn tend to have good deals too. Both Chain and independent hotels in the city centre tend to have smaller than average rooms due to the fact they are in very old buildings, the Marriott is out near the zoo and has lovely big rooms but you do need to get a cab into the city centre.
The museums are free and you can look round the Scottish parliament building or go on a tour for free too. (not too interesting though) The parliament I mean, the museums are brilliant. If you like history Hollyrood Palace is beautiful and steeped in history, the guides are excellent, tours can be restricted if there is a member of the Royals in residence although they mostly stay at Balmoral. Mary Kings Close tour is fun, you go under the buildings on the Royal Mile where The old town has been preserved and find out about how people lived 100s of years ago. A night time ghost tour is fun but does require a good bit of walking on cobbles in some cases.
My favourite places include the museum of childhood, the writers museum, and Edinburgh's camera obscura, the actual bits on the way up to the tower are aimed a children but once you're up at the top the views are stunning and the camera obscura is fascinating. You have the best effect if it's a clear day, also it's a genuine Victorian obscura and of course the Castle is a must.
If you like fine dining and would like the Scottish experience The Witchery is a must but it comes at a price, nice for a vacation treat but there are many lovely more affordable restaurants on George street.
If your a confident driver you can hire a car and head over into Fife, StAndrews and the east nuke coast with its little fishing villages are nice, or up to Perth And Pitlochry where the scenery becomes more rugged. There are private tour companies that will drive you to the places you would like to see but I'm not sure of the cost. Stirling Castle is also nice
If you would like any questions answered, ask away. :)