absolutely what Steve said...do NOT take the Leonardo Express to Rome(that puts you at Termini Station which is across the city from where you want to be. Take the regular train to Trastevere Station..it is where you want to go and is much cheaper(around 4 euros or so)
From Trastevere Station you can take a cab or hop on the tram right outside the station to Trastevere proper.

At SlowTravel, Alice Twain wrote a wonderfully detailed description of how to use trains in Italy. Here's the link: www.slowtrav.com/italy/trains/index.htm

We buy our train tickets at a travel agent's office. We usually pass one in the course of our touristing. Just look for a travel agency with the train symbol in the window. Once we left the long lines and pandemonium in the Stazione Termini (the big station in Rome) and went to a travel agent inside the station. It was calm and quiet and there was only one person in line in front of us.

Mimar's link is very nice and informative.

There is also a video here of how to use the electronic machines, which even in Termini will help you avoid the lines:

http://www.webvisionitaly.com/catego...p;ref_item=375

The local train to Trastevere station described by Steve James and russwuf is definitely better --both more direct and cheaper-- than the Leonardo express for your purposes in getting from the airport. It leaves from the track opposite the L Express at the airport. It was marked to the destination Fara Sabina rather than as F1 when we took it

Tram #8 will take you from the Trastevere station up Viale di Trastevere. You catch it out at the main road,not in the plaza area right in front of the station. The stop is slightly to your left as you leave the plaza. From there it will turn the corner to head north up Viale di Trastevere

The same tram #8 continues north of Trastevere across the Tiber to the Largo Argentina which is a central transit hub

We found the automatic machines for trains to be unreliable sometimes and the automatic machines for buses/trams in Rome to be unreliable 99.9% of the time. Better to buy either a multi day transit pass in Rome or individual tickets at tobacconists. The Roma Pass does include some transit ticket options

You didn't ask about advice on time allocation for the various destinations but I strongly recommend spending at least four full days (after the first jet lagged day of arrival) in Rome. Shorter stays in Rome tend to be really frustrating.