The easiest way to get into the city by public transportation is to take the Termini Express train from the airport. Follow the signs for FS (Railroad in Italian) from the terminal. You can buy the tickets either from the automatic machines or from the ticket booth. Normally it leaves from the track on the far right. It goes non-stop to Termini Station. Then you can get the Red Line (Line A) to Barberini (two stops). This takes you to Via Veneto. Walk up the slight hill and around the bend past the US Embassy and you are there.
The alternative is to take the local train to Ostiense and then get the subway there but you would have to transfer from Line B to Line A at Termini. It is a bit less expensive though.
Sometimes taxis can fit six people. They do have some taxis that are like small mini-vans. Of course it is Italy so laws are made to be broken though not everybody is willing to chance it.