The heart of tourist Rome is the Via Veneto, in the north-central part of the city. I've never figured out why it is the heart of tourist Rome (most of the buildings date from the late 19th and early 20th centuries), but if you feel like trying (and can afford340,000 lire per night) stay at the Regina Baglioni, via Vittorio Veneto 72, 00187 Roma, tel (06) 476851, fax 485483. The suites on the 7th floor have a beautiful view of the city and, especially for Americans, there are some non-smoking rooms. You'll be close to the important museums and beautiful gardens in the Villa Borghese and to the Roman Sports Center, the only athletic club in Rome that admits non-members so far as I know. Finally, if you lose your passport while staying at the Regina, you can walk across the street to the American embassy and get a new one.
If you want to be closer to medeival Rome, including Trastevere, check out the Campo de' Fiori hotel, Via del Biscione 6, 00186 Roma, tel (06) 654 08 65, fax 687 60 03. Prices are medium (perhaps 150,000 lire per night) and rooms are small, but you're two steps from the open-air market in Campo de' Fiori, about a ten minute walk from what passes for Jewish Rome, and mere minutes from Trastevere, which is a hotel desert.