The one unique thing is Holyrood Park, going up to Arthur’s Seat - an extinct volcano in the middle of a city. Really extraordinary. Lovely strolling on the hills with great views in every direction.
There are plenty of guided walks and tours, particularly around the Royal Mile, but I’d say what makes Edinburgh special is that it does normal capital city things spectacularly well.
The cityscapes are amazing - and you may have bright blue skies through December making everything even more glorious. Just stand anywhere in the Old or New Town and turn 360 degrees - it’s impossible not to be awestruck - so factor in a lot of leisurely walking and stopping for coffee in any of the countless, peculiarly cosy coffee shops.
Galleries - The Scottish National Gallery of Modern Art, split between two buildings on either side of a road, is probably my favourite place in Edinburgh. The outside space, with lawns dotted with sculptures, a wooded walk down to the Water of Leith, and views over the city, and the inside spaces, are outstandingly lovely. Lots of other ‘National’ and other galleries, too - all impressive. And the grandest, most absorbing museums.
Try and find a play to go to, maybe at the Traverse theatre, or a classical concert.
Not sure of the food market situation right now. Worth finding out - I remember the farmers’ market beneath the castle being particularly fun.
As for eating - find a place that boasts of a view of the Firth of Forth. Harvey Nichols restaurant used to be good for this - no idea if it’s still going.
(I love Edinburgh in December - you’ll have a magical time.)