We go to Edinburgh every year for a few days. It's a lovely city.
I would recommend the National Museum of Scotland, the Writers Museum (very small, commemorates Robbie Burns, Walter Scott and R L Stevenson, if I remember rightly), Dynamic Earth, Holyrood Park and Arthur's Seat, and a tour of Mary King's Close. My family have done a couple of the walking tours, the ghost ones, and said they were brilliant. I haven't done them because I have bad arthritis and wouldn't be able to keep up. As an Ian Rankin fan, I think that the Rebus tour would be good, but again I think that you need to be able to walk at a reasonable pace.
My friend toured the Britannia and said it was well worth the money but it has never appealed to me.
The Botanical Gardens are very good, if the weather is kind to you.