Calton Hill is my favorite spot in Edinburgh. It has an excellent view of all the highlights of the city: the castle, Arthur's Seat, all the spire tops and domes, the Forth River, Holyrood Palace... If you haven't got much time in Edinburgh, you must have a quick climb (there's stairs, and it takes about 5 minutes, so it's not really much of a climb) up to the hill to check Edinburgh in all of it's glory.
There are also loads of monuments, such as the "National Monument", as pictured above. (Notice the resemblance to the Parthenon? It was built in it's likeness). It is known as "Scotland's Disgrace" because it was intended to be Scotland's national memorial to the Scottish soldiers and sailors who died fighting in the Napoleonic Wars, but was left unfinished due to financial hardship. Still, tourists and locals climb atop the Monument (with a running start) and enjoy it regardless.