Take the bus at the airport to AGNANINA subway. Red line into city.
For less than 2 euros you'll be on the red line into Rome center [centro storico], to Termini Station (1 euro bus, 1 euro subway).
The red line goes on to Spanish Steps and Piazza del Popolo.
From Termini Station you can take the #40 or #64 bus, in front of the station to the Largo Argentina area, Rome hotels, nearby are the Pantheon, Campo de' Fiori, and Piazza Navona Centro Storico Rome.
Enjoy Rome at a glance.