Airports
There are 2 airports:

Fiumicino is the main international airport and is situated to the west of Rome. Taxis to and from Fiumicino offer a fixed rate for up to 4 people and their luggage to anywhere within the old city walls of 40 Euro, if it is outside the old city walls then the meter applies. The train is expensive for Italy costing 11 Euro single to Termini. There are other trains to Rome Tiburtina that costs 5.50 Euro, these also stop at Trastavere and Ostiense so can be an option if staying near to there. There are some buses but they can only be used if you are traveling with certain airlines for some reason, there are also local buses but they are infrequent.
Ciampino is where most of the cheap airlines fly in to and is situated to the south of Rome. Taxis to and from Ciampino offer a fixed rate for up to 4 people and their luggage to anywhere within the old city walls of 30 Euro if it is outside the old city walls then the meter applies.
There is no direct train service in to Rome however there is a regular bus service from the airport to either Ciampino station or Anagnina A line metro station. The ticket costs 1 Euro and can be bought on the bus, the bus stops are straight in front of you as you exit the arrivals hall. The train from Ciampoino station costs 1.30 Euro to Termini. The metro is 1 Euro for a journey for up to 75 minutes in the same direction, this means you can change to a bus or the B line metro with the same ticket. The Terravision bus is expensive at 14 Euro return, this takes you to the Termini station so you still need to get the bus/metro or a Taxi from there. This is the bus that Ryanair and Easyjet sell on their flights, you can also buy a ticket in the arrivals hall. There is also a bus run by the national bus company (Cotral) that costs 5 Euro that again takes you to Termini station, you can buy a ticket in the arrivals hall.

For more information on the airports visit www.adr.it