We had a fantastic three night stay in Agata's flat in Trastevere. Silvia and Agata were excellent hosts, who were always available to help us (not that we needed it), gave great restaurant recommendations, and even left us a bottle of Prosecco in the fridge to enjoy!
The flat itself was perfectly located in buzzy Trastevere; the street was lined with lovely restaurants and fun bars (but not at all noisy) if you want something close to home. Otherwise all the main sites are no more than a 15 minute walk, easily accessible. Campo di Fiori is a ten minute walk away and a perfect place to start your day with some breakfast and a walk around the market.
I could not recommend Agata's flat any more; your experience of Rome will be all the better for staying there.