This turned into a much longer post than expected so I hope it's helpful

I absolutely love Edinburgh and it will always have a special place in my heart. I have family that live in Edinburgh and used to split my summers between there and Glasgow (my mom is from Glasgow). When my wife and I move in a few years, it will probably be within an hour of Edinburgh. Right now the plan is to spend a month or two there this summer. I won't say I'm an expert, but I know a few things.

Some of my favorite places:

• Edinburgh Castle-I feel like this is a must. The tour is full of so much history and its actually a really cool castle as it isn't in ruins.
• Mary King's Close-One of the older preserved parts of the city. The Old Town is literally built on top of older parts of the city. Mary King's Close was once an open air street that now sits under the city.
• Victoria Street -Victoria Street has lots of little shops and is the inspiration for Diagon Alley in Harry Potter. There's a shop called Oink that sells great pork sandwiches and Walker Slater is a good option for higher quality clothing than what you would find in the normal tourist shops. FYI Victoria runs into West Bow street, they are effectively the same if you are looking for a specific address
• Grassmarket- Square with shops at the bottom of Victoria, there is a has a pub called The Last Drop. Grassmarket was once the place for public executions and the condemned would be offered their last drink in the pub before their date with the gallows.
• Cadies and Witchery Tour-This is a night time walking tour that takes you to all sorts of places from the City's dark past. This is the best of the haunted type tours I've done in the city. The main location is on W Bow/Victoria and they have a card holder made from the skin of William Burke of the infamous Burke and Hare body snatching duo. They also have death masks of both William Burke and William Hare.
• Royal Yacht Britannia-Actually one of my favorite things I've toured. It's a prime example of 20th century Royal Monarchy style and opulence. It has sailed the royals all over the world and is in excellent condition. Lots of cool memorabilia and interesting to see how they traveled during the 50s-80s
• National Museum of Scotland-Fantastic Museum that's good for young kids as well, plus it's free. Tons of Scotland's history is located here including 11 of the Lewis Chessmen. From the 12th century, they were discovered on the Isle of Lewis. They are some of the oldest chess pieces in the world and actually look like human figures.
• Signet Library-Home to the Society of Writers to His Majesty's Signet. Dating back to the 15th century, they were authorized to produce royal manuscripts with the king's Signet. More importantly, do you want to feel fancy? I mean really ****ing fancy. The kind of fancy that would throw gold coins from your majestic carriage at the beggars in the street as you make your way to a ball. The Signet Library has an afternoon High Tea that is straight out of Bridgerton. Sweet and savory amuse bouche, champagne, ros, cocktails, tea, coffee, etc. My favorite tea is the Irn-Bru pot. It's based of the national drink of Scotland and has a fruity orange flavor.
• SCOTCH-This is the Whisky bar inside of the Balmoral Hotel. Skip the Scotch Whisky experience and go here instead. They have an enormous collection of whiskys including lots of rare bottlings. Whiskys are served in drams so a little less than a shot. The upside is that you can sample some really rare whiskys you wouldn't likely every see or afford. The experts here are basically sommeliers of whisky and have tried all 500 plus offerings on hand (they all wear kilts too). It's a great place to unwind after a day of sightseeing with a nice fireplace and comfortable leather couches. If you see Frasier, tell him Ian from Texas says hi (he'll remember me because I'm the only one who has ever requested his least favorite whisky).
• Holyroodhouse-Official Residence of the Monarchy of Scotland, its where the King will stay when he visits. Part palace and part museum, it has a lot of artifacts from Scotland's monarchy and history. At the end of the Royal Mile, its a worth the visit. Lots of history on Bonnie Prince Charlie as well.
• Greyfriars Kirkyard and Bobby- Very cool old cemetery. There are tons of walking tours, but you can do it yourself with what's available online. Several of the characters from Harry Potter were pulled from the graves here. Just outside is the Greyfriars Bobby Statue. It's a real life Futurama Jurassic Bark episode. The story goes that the dog's owner died and was buried in the Kirkyard. The dog sat by his owner's grave until he died 14 years later. The dog is buried near his owner and a bronze statue was dedicated to him just outside. So many people have rubbed his nose for good luck that it's now shiny and deteriorating. If you must do so, just give it a gentle tap.
• Calton Hill-I don't know how to describe this place other than the monument distinct (Like the Simpsons hammock district). There are a bunch of different monuments and the only real connection between them is that they are all Scottish. That aside, it has beautiful views of the city especially at sunrise and sunset. A small word of warning, it's not a place to visit after dark. It has long been a magnet for what the Scots call "antisocial behaviour", prostitution, and dogging (public sex). Or you know, you do you if that's your thing.

I hope this is helpful. If you want recommendations on places to stay, eat, or shop let me know and I can put something together for that as well. I can also mention some specific seasonal events if you like