well this is probably repeating a lot of what people already said but:

Sights
—Climb (meaning walk up some stairs) to the top of Calton Hill. it’s a great view of the city and beautiful in of itself.
—Arthur’s Seat (likewise)
—Prince’s Street Gardens and the fountain (there’s a huge beautiful fountain with the castle as the backdrop). there’s also usually some kind of street market or fair taking place across the street in new town near that end on the weekends.
—Cockburn Street(right off the Royal Mile) is a little winding cobbled street. there are some funky shops, a good Tapas Restaurant and a great view of new town at the end of the street. Make sure to make your way down some of the closes that wind off of the street. go down the Royal Mile too:0)

Eating:
these places don’t necessarily count as “restaurants” but they are some great places to find good foood/respite from what can be some gloomy weather.

Chocolate Soup—right off the royal mile. they serve hot chocolate, coffee, expresso very elaborately and beautifully, often, in a bowl!

Central Mosque Kitchen: there’s a mosque located next to the university in Old Town (I think it’s at West Nicholson Street and Potterrow…look for the Mosque). There are meat and veg dishes served in a big bowl, and you can sit under a big tent. It costs 3 or 4 pounds and was one of the best hot meals I had in the city. It’s a great quick lunch place…but they’re not open on Sundays.

Brass Monkey: one of my fav. spots in the city is located on Drummond Street in Old Town. It’s a pub that serves coffee and sandwiches during the day that happen to be delicious. Brass Monkey’s got a Moroccan theme and one of the rooms is outfitted with a gigantic bed/sit on cushions with miniature tables for pints or coffee. At 3pm every day the bar plays a movie and you can sit and relax on the gigantic bed.

other restaurants—there are a lot of great little places in the Grassmarket Area.

Have fun! Edinburgh is a beautiful city.