Rome’s central bus square (for the local buses) is the Piazza dei Cinquecento. It is situated in fromt of the main train station Roma Termini and from here almost any destination in Rome can be reached easily.
There is a unified ticket system for local buses, trams, the underground and (some) local trains.
For interlocal buses there are several other squares from where you can leave, while the buses from the two airports also arrive and leave from Termini.
The international coach company Eurolines has its office and main stop on the Piazzale Tiburtina, near Rome’s second railway station, also called Tiburtina. Tiburtina Station is on line B of the Roman underground system.