We stayed in an apartment in the Monti neighborhood, close to the ancient sites. The city was very accessible from there, with a Metro station (Cavour) just blocks away. I'd also recommend spending time at Villa Borghese.
The main reason we were there, though, was to run the Rome Marathon, which gave us an extraordinary tour of the city through the streets, which were blocked for the runners.
We spent a week there, but are planning on returning, and skipping the big attractions. Staying in Monti gave us a fantastic view of the Colosseum many times a day, and it never disappointed. It was an odd sensation of how familiar it became as we went about the city.
It boggles to consider how much is *left* and visible, considering the amount that was rendered into lime mortar during the Middle Ages, demolished even during imperial times, and reused or left buried over the centuries. :^)