Mary Kings Close – book beforehand.
Museum as noted above – there are 2 next door to each other.
A walk around Greyfriars Kirkyard and see the elaborate mausoleums, and read of the awful history of the place.
The National Gallery is one of my favourites. It is undergoing refurb (finished yet?), so only half was open last time I was there in August, but still worth a visit.
The Portrait gallery was refurb’d a coupe of years ago, and is a beautiful building, with some really good exhibits.
Take a climb up the Scott Monument, laugh at fat Americans trying to get up the narrow staircase, take a camera to take pics from the top.
Want a brisk workout, walk up to Arthurs Seat, plan on being half a day doing that. Or, take a taxi up, and walk down.