Have you read a decent guidebook on Rome yet? (There's this Rick Steves guy who does pretty good ones.) That will give you a lot of information you might not think to ask. For example, your follow up post seems to imply that you think the historic center of Rome may be very large and that you need to stay outside it and take the Metro in. That's not true. The historic center is 2.5 miles east-west and 1.5 north-south. There are LOTS of quiet options within it.
Also, access to the Metro isn't as important in Rome as in Paris. Look on Google maps to see where the Metro stops are and you will see that it forms a circle around most of the historic center and doesn't go into it.
Also, start thinking in euros, not USD. For apartments, we've had good luck through Airbnb, vrbo and booking.com. Choose only places with a lot of reviews including recent ones. For a quiet area, consider the Prati neighborhood.
Happy travels.