The most charming, historical area is around the Pantheon. Piazza Navona, and Campo dei Fiori. Anywhere in those areas will be fine, as long as it isn't on Corso Vittorio Emanuelle (too busy of a street). The downside is they are not right near a metro stop - but the metro doesn't go through the historical area, it skirts it. You will be walking a lot anyway!
Where you stay in Rome depends on both your budget and how convenient you want your lodging placed. Centro Storico is the area around the Pantheon. Campo Fiori, and Piazza Navona. It is considered the center of modern Rome and everyone likes/wants to stay there because you can walk to a great number of Rome sites. Because of its popularity, lodging in Centro Storico command a premium to other areas. Same with the Spanish steps. You have to balance your budget restrictions with your convenience factor. Rome is a big place and unless you want to spend 30 min or more walking, there are sights that it would be best to take public transport to, even if you base yourself in Centro Storico.
Trastevere, the areas around the Colosseum (Monti, Celio), Barberini area, Vatican are not as central but reasonably charming and convenient. Using Public transport (reasonable good and cheap in Rome), these areas are reasonably convenient. Trastevere is my personal favorite. There are great transportation connections in Trastevere because
Tram 8 that services the area runs every 15 min.
At the other end of the spectrum is the area around Termini. This area gets a bad rep, but violent crime in Rome is very low by American standards, and I have stayed in this neighborhood overnight and walked around there more than once after dark with no bad effects. Lodging near here is some of the cheapest in central Rome, but you have to take public transport to nearly all the major sites it you don’t feel like walking 1-2 miles to get there. The other down side is that is charmless and modern
Distance from Termini
1.45 miles to Piazza Navona (as the bird flies)
1.27 miles to the Pantheon (2.04 km)
1 mile to the Colosseum
2.5 miles to Vatican city (St Peter’s Basilica)
Bus 40 into Centro Storico
The Rome Metro is not as comprehensive as in other European cities like Paris. There are only 2 lines that cross at temini currently (feb 2011). Someday there will be a third line in Rome which at this time is hopelessly behind schedule. While the Metro is a valuable addition to transport options, to get to many parts or Rome unless you want to do a lot of walking, you will need to use the bus system.
Map of the two Metro lines and the regional trains