Avoid the Termini area. It's the area around the train station. Areas around Trevi fountain, Pantheon and the Spanish Steps would be all nice and central.
There is a hotel called Hotel San Francesco in the Trastevere area which I would
reccommend. We stayed near Termini Stn a few weeks ago and friends of ours
stayed in this hotel and it was a lovely area. Near some nice Piazza's with nice
bars/restaurants and there is a lovely rooftop bar in the hotel itself.
You can book on http://www.venere.com (www.venere.com) which is a great site.
Also would recomend doing the open top bus tour of Rome. 11 stops to all the
most popular sites for E16 and you can get on/get off as you please. Really
nice way to see the city.