When I was in Edinburgh I enjoyed hiking up Arthur's Seat for the view and the ruins of St. Anthony's Chapel.

I also liked seeing both ends of Hadrian's Wall, at Wallsend on the east and Birdoswald on the west. Both sites have Roman fort ruins and museums.

As for non-super-touristy (well, I guess it's still touristy but more quirky) I really liked watching the Millenium Bridge in Newcastle tilt to let river traffic go under!