My home town! It's really a beautiful place.
Depending upon when you're there, you should still be able to take the usual open top bus tour which I would recommend. That will take you to all the main things to see - Edinburgh Castle, Holyrood Palace, Scottish Parliament, New Town, Royal Mile, Grassmarket etc.
There's also Mary Kings Close which is the underground city. If you're into ghost walks, then there is plenty of them all leaving from the Royal Mile.
The Botanic Gardens are worth a visit as is the Shore down in Leith - the Royal Yacht Britannia is in this area as well as some really lovely restaurants.
If you have children with you, then perhaps a trip to Our Dynamic Earth? This is really close to the scottish parliament.
A picnic in Princes St gardens is a great thing to do if you like to people watch in a lovely calm and peaceful setting.
There's no limit to pubs and restaurants for drinks and meals.
There's also quite a lot of art gallerys - on Queen St, Belford Road, Princes St
A friend of mine from Canada was completely obsessed by the Forth Rail and Road bridges - not my thing, but you never know!
That's a good start