Must sees in Edinburgh

I would say Calton Hill isn't mentioned enough for its beautiful views of the city. Holyrood House is pretty special, and it doesn't always get much love from tourism types. Otherwise the main widely-advertised attractions seldom disappoint.

Arthur’s Seat

Dean Village is lovely. It feels off the beaten path, but is actually an easy walk from the west end of Princess Street/New Town area.

To save time I've cut and pasted an excerpt from my Trip Report from a couple of years ago.

The gem of the day was the small museum inside the National Library of Scotland which we found by accident. We were walking toward the National Museum of Scotland and saw the National Library and decided to pop in because, a: I love libraries, and b: libraries usually have Wi-Fi; and it did. Inside the main doors and to our right was a small museum with rare books and artifacts that we loved https://www.nls.uk/treasures/. One that really drew our attention was a display about the Battle of Culloden. The display included a hand drawn map by a French Officer was at the battle, and also a deck of playing cards that a Jacobite Officer used to write his diary on while he was on the run while protecting Prince Charlie. It only takes about half an hour to walk through the museum

I would agree with both suggestions of Calton Hill and Holyrood. One of my best photos of my trip was taken from Calton Hill. I also visited the Victorian manor house which gives a great insight into Victorian life if that sort of thing interests you.

I've always enjoyed Gladstone's Land on the Royal Mile which was the home of a wealthy C17th merchant. Read more about it here.

Completely different is the Georgian House in Edinburgh New Town which is C18th/19th

Scottish National Portrait Gallery

The National Museum of Scotland …..SO worth it and good views from the rooftop terrace.
The Museum of Childhood on the lower part of the Royal Mile.
Get on a double decker bus and ride along looking from above.

https://www.lothianbuses.com/

The Royal Yacht Britannia.
The Botanic Gardens.

I agree with SJ about the National Museum of Scotland and the Botanical Gardens! The Museum is soooo interesting and quirky. It's the final resting place of Dolly the cloned sheep! And the botanical gardens are gorgeous. First time I ever saw blue poppies! I highly recommend a guided tour of the Royal Mile and Edinburgh Castle with Mercat Tours.

And perhaps a night time after dark ghost tour of the Royal Mile.
Corny, but fun…pick one that stops in a pub as a group at the end.

Second most of the above. Used to go to the galleries, both the portrait and national, and sometimes just sit and make up stories about them. The NMS is great in Edinburgh, it also has Scotland's proto guillotine.

For the ghost tours, I did one in York and it was excellent and Edinburgh has a similar haunted reputation. Best thing though is check the reputation of any you see, you want a good entertainer/raconteur as much as anything. They can bring these stories to life, so to speak.