Rome has two airports: Rome-Fiumicino (also called "Leonardo da Vinci", FCO) and Rome-Ciampino (also "Giovan Battista Pastine", CIA)
Rome-Fiumicino (FCO) is by far the bigger airport and 20 miles away from the city.
If you prefer to take the train you can either take the "Leonardo Express" or the trains of the line FR1. Leonardo Express will take you to Stazione Termini in about 35 mins for 14 € per person. The trains go every 30 mins. If you need to go to Trastevere, the FR1 line is the better choice, getting you there for 8 € (but not to Termini).
The budget alternative are the Terravision buses that get you to Termini Station for only 5.80 € but take almost an hour for it.
A taxi ride from FCO to the city costs a flat fee of 48 €.
The Car-Sharing service ShareNow (formerly Car2Go or DriveNow) is operational in Rome and recently there are also cars available at the airport (Terminal 1). For journeys from the airport to the city an extra 10 € is charged. Furthermore, a ShareNow car is one of the best ways to tackle the last mile to your hotel or apartment, e.g. if you arrive by train in Termini!
Rome-Ciampino (CIA) is nearer to the city (10 miles) but not a lot of airlines fly to CIA.
The Terravision buses get you to Termini Station in about 45 minutes for 5.80 €. The interval is 20 to 45 mins. There are other bus companies as wells, such as CoTral, Atral or Schiaffini, that offer basically the same.
Taking a taxi from CIA to the city will cost you a flat fee of 30 €.