Calton Hill, a ten-minute walk east of Princes Street, is the best viewpoint in Edinburgh. The hill is low enough to climb without serious effort and high enough to give a panorama of the whole city: the castle on its rock to the west, the New Town grid below you, the Firth of Forth and the hills of Fife to the north, Arthur's Seat and the Salisbury Crags to the south. The view is especially good at dusk.
The summit is cluttered with monuments in a very Edinburgh way: the unfinished National Monument, the Nelson Monument (a tower you can climb for an even better view), the City Observatory, the Dugald Stewart Monument and several others. The whole ensemble has a pleasantly eccentric, slightly ruinous character that suits the city. The hill is freely accessible at all hours.