Edinburgh is kind of divided between what's known as "Old Town' & 'New Town'. Old Town basically sits above the New Town area and that is where you'll find the Royal Mile and Edinburgh Castle. The Castle is at the top of the Royal Mile and Holyrood Palace sits at the bottom. Along the way are many shops, pubs, restaurants and interesting sites. It's best to find a ride to the castle (Taxi, Hop-on/Hop-off tour bus or City Bus (Lothian Buses) and walk down- much easier as getting to the castle on foot is a pretty good climb.
Definitely stay in the City Center of Edinburgh. The hotels there will be a bit more expensive, but well worth it in my opinion. Anywhere in the 'West End' or York Place/St James Quarter of New Town will be great as the Edinburgh Trams run pretty much everywhere in the City Center, even down to the Leith area (seaport region where the Royal Yacht Britannia is moored).
There are also a number of nice hotels along the Royal Mile, as well as the Grass Market which is just below the castle; however, Trams do not run in those areas, so you'll have to walk or grab a Taxi or Bus. One positive thing about the Edinburgh Buses (Red Lothian Buses) is the fact that they all run down Prince's Street at some point- Just make sure to ask the driver if he is headed to or away from Princes Street.
The Royal Mile and Edinburgh Castle are pretty much a 'must see', they are a pretty tough climb from Prince's Street which is the main shopping street. The Edinburgh 'Hop-on/Hop-off' buses are a great way to get a general overview of the city and also will eliminate some of the walking/climbing between the main sites. I would do that, stopping off along the way to see the Castle and walk DOWN the Royal Mile.
Depending upon how much time you have, some of my favorite places are as follows:
The Surgeon's Hall Museum on South Bridge/Nicolson St
Scottish National Gallery on Prince's Street
National Museum of Scotland (near the University)
The Edinburgh Playhouse (There is usually a good play on)
The Conan Doyle Pub for a meal (named in honor of Arthur Conan Doyle, the creator of Sherlock Holmes). Conan Doyle lived nearby.
Stockbridge area if you have time and check out the 'Charity Shops'- Fun & full of bargains
The basement supermarket at Mark's & Spencer's on Prince's Street.
Primark Clothing Store- Prince's Street
Walk along Rose Street which is like a pedestrian only street right behind Princess Street.
TISO outfitters on Rose Street
Stop at the Gregg's bakery on Rose street for a 'Steak bake'
Also, any of the 'Pret' sandwich shops for a delicious quick snack
Edinburgh is a magnificent city and, in my opinion, one of the World's most beautiful. It's easy to get around and people are friendly- even if a bit hard to understand at first. You'll love it!