If you love Harry Potter, don't bother with the tourist trap shops. Instead take yourself to Greyfriars Kirkyard where JKR found inspiration for names of characters on the gravestones- she wrote the first HP in the Elephant House Cafe which used to be just in front of Greyfriars, but it's currently closed due to fire damage. There are some fantastic tours of the Kirkyard, some with an HP theme. Loads of other fantastic stuff to look out for there too - mortsafes from the days of the resurrection men, the Covenanters' Prison and the overall atmosphere of the place. I love it there.
Mary King's Close is good but I prefer the Mercat vaults tours of the vaults under South Bridge. Incredible and heartbreaking that people used to live in them.
The Museum of Scotland is fantastic, you could spend the whole day there and still not see everything. The Scottish galleries are fabulous, and will take you from prehistoric Scotland in the basement right up to the present day on the top floor. The view from the viewing terrace is breathtaking.
A walk up the Royal Mile from Holyrood to the castle is a good shout. Explore the closes and wynds off the main street, lots of good places to stop off for lunch (the haggis pasties at Mòr Bakehouse on Hunter Square are to die for and they're so huge you may well). The castle is one of my favourite places in the world - try and get there for the one o'clock gun and don't miss the Honours of Scotland (the Scottish crown jewels).
The National Gallery of Scotland is wonderful and the gift shop is especially good, as is the new café.
Surgeons' Hall Museum is fascinating, shocking and humbling in equal measure.
The Botanics are gorgeous, particularly at this time of year (a wander through Stockbridge on the way is always nice). In fact autumn is imo the best time to see Edinburgh full stop - you're very lucky! I used to live there and get back whenever I can.
My best advice is to read about Edinburgh before you go - the history is fascinating and you'll get right under the skin of it as it were.
Have fun OP!