This article outlines the various ways of getting from and to Rome Fiumicino (FCO) airport. The various options are listed, approximately, from least to most expensive. The cost of private car services varies depending on number in the party and other factors. The more expensive options are often also more convenient, although convenience depends also on the location of your hotel. It's a good idea to get a map of Rome and become familiar with the location of your hotel and of the places mentioned in this article.
Buses
Roma Airport Bus 
Check www.romeairportbus.com for a company called Schiaffini offering one way tickets from around 4 euros with 23 trips to FCO. They are situated in Via Giolitti at Stazione Termini. Info line +39067130531.
T.A.M. (€5 per person online or on the bus).  Between 08:00 and 23:30, this shuttle service operates between FCO (Terminal 3) to Termini Station. The buses are located at Terminal 3, and ticket price is €5.00 per person online or in person.  Open return tickets cost €8.00 per person. From Termini Station (on right side as you are facing the station) they travel between 04:30 and 20:30.  See the timetable.  They also stop in the center of Fiumicino town, and at Piazzale 12 Ottobre, near Ostiense train station, where you can connect to Italo high-speed trains to Florence and points north.
Terravision Shuttles (€4 per person online, €6 at the booth). Between 05:35 and 23:00, this shuttle service operates between FCO (Terminal 3) to Termini Station (Via Marsala). See www.terravision.eu/rome_fiumicino.html . Be aware of the fact that sometimes the bus is full ...
SIT Airport Bus (€5 per person online, €6 at bus station). SIT run more than 20 buses daily in each direction. They run between 05.00 and 20.30 from Rome, and between 08.30 and 00.30 from FCO. The stops in central Rome are at Piazza Indipendenza (near Termini station) and Via Crescenzio (near Castel Sant'Angelo). The journey between FCO and Termini takes 50 minutes. Seehttp://www.sitbusshuttle.it/index.php...  If you are catching the SIT bus do arrive at your stop at least 20 minutes before the scheduled departure; buses can arrive - and leave - early. At the airport, the pick up and drop- off points are outside the arrivals hall at Terminal 3.  This is a considerable distance from Terminals 1 and 2, where all domestic and inter-Schengen flights arrive and depart.
Cotral night bus from Tiburtina and Termini (€5 if bought in advance)  For those who have to reach Fiumicino airport at a very early hour, before the trains and other buses are running, there is a night bus that connects central Rome to Fiumicino airport for a one-way cost of €5. See the schedule athttp://www.cotralspa.it/PDF_Areoporti... . To avoid paying more for the ticket on the bus, you should buy the ticket the day before at one of the sales points mentioned on the schedule. At the same time, you should locate the bus stop so that you don't have to search for it in the dark with your luggage. 
Cotral bus plus train or metro (about €3 per person).  You can also get to central Rome using only public transportation. Although it would cost only a little less than the Terravision shuttle, and would take longer if you were going to Termini station, it might be more convenient for people going to some parts of the city. You can take a Cotral bus from stops outside Terminal 2 or Terminal 3 to Ostia Lido, and take the Roma-Lido railway from there to Ostiense station, followed by Metro or bus to your final destination; the Cotral bus costs a few euros and the Roma-Lido railway can be ridden with the same bus-tram-metro ticket used by the rest of the ATAC (Rome's public transportation agency) network, which costs €1.50. You could also take a Cotral bus to the Anagnina Metro station, and take the A line from there. This route would be somewhat longer, but could be more convenient if your destination is on line A. The cost would be about the same. The tickets for the Cotral bus, as well as for the metro or a bus, can be bought at a tobacco shop or newsstand. If you buy a a BIRG ticket (€8), you can use it for the Cotral bus and for all metros, buses, trams, and urban trains (including the Lido train) for the rest of the day. This route isn't recommended for people with a lot of luggage.

Suburban train (€8 per person).  The FL1 suburban service has a number of stops: those close to the centre of Rome are at Trastevere, Ostiense, Tuscolana, and Tiburtina stations. At Ostiense station, this service connects with metro line B, and also with high-speed Italo trains to Florence and points north.  At Tiburtina station, the FL1 connects with Metro line B and also to Italo and Trenitalia trains. At Trastevere, it connects with tram (streetcar) line #8, which runs along viale Trastevere (the main street of Trastevere) to piazza Venezia: from the second-to-last stop (the "Arenula/Cairoli" stop), it's a short walk to Campo de' Fiori, Piazza Navona, and the Pantheon, so this route may be helpful if you're going to that area and do not have an excessive amount of luggage. At the Tuscolana station you would have to walk a few block West to get Metroline A at Ponte Lungo Station. See all stops.
Express train (€14 per person). The Leonardo Express train is a non-stop service between FCO and Rome Termini (the main station). It is very useful if your hotel is near Termini or if you want to catch a train to some other part of Italy. Many trains to Florence and points north also stop at Tiburtina station, which is reached by the less expensive FL1 train.
Taxi (€48 per taxi). The fixed fare for a Rome taxi is €48 between FCO and central Rome (within the Aurelian walls) for up to four people and their luggage. If your hotel is outside this area, you will have to pay whatever is on the meter. For a map of the area that's covered by fixed rate fares from the airports, and the current rates, see the Rome taxi fare schedule, in several languages, with a map of the fixed rate area on the last page:http://www.comune.roma.it/PCR/resourc.... 
For other journeys, ask around to see what the taxi should cost or get an estimate online. Have small bills to count out carefully when paying. If possible, snap a quick photo of the tag number on back of the taxi before you get in. Then you have a way to trace them if there are problems. 
Note that the fixed price from the airport to central Rome applies only to taxis licensed by the City of Rome; these taxis have the City Council's crest (with "SPQR") painted on the doors. Other taxis, such as those licensed by theComune of Fiumicino, can charge by their meter. Insist on taking a Roman taxi: they now have the fixed rate to and from the airport (€48) printed on the doors. 
If you are travelling as a group of 4, a taxi is cheaper than the Leonardo Express train and avoids having to lug your suitcases to the train and then having to make connections on other transportation once you get to central Rome.
Private car and driver (€50 and up per car). For perhaps less money than a taxi, you can have a driver waiting at the exit from Customs at FCO or at your door in Rome. Prices start at €50 for three people.  Frequently recommended on TA are  www.romecabs.com and http://www.romeshuttlelimousine.com/. Private car services usually charge more if there are more than two or three people in the group, so the service may or may not be cheaper than the train. Private services also usually charge more for early morning or late evening transfers, so you need to get a quotation for your particular situation. For up to three people, during the day, a private driver is often cheaper than a taxi. 
Avoid any car or shuttle service that requires payment in advance. The reliable companies ask for payment after they have provided the service.
Tipping: It is never necessary to tip for anything in Italy, although equally it is fine to give a small tip (no more than 10%) if you feel you have received excellent service. You are not obliged to give a tip to your car or taxi driver.