Trenitalia has two direct Frecciargento high-speed trains from FCO airport to Florence, Bologna and Venice. These two trains (at 11:08 and 15:08) pass through Roma Termini and Roma Tiburtina (to pick up more passengers), but you stay on the train all the way to your destination. It's usually not a good idea to book these trains in advance, since you run the risk of forfeiting your ticket if you miss the train because of a late-arriving flight (I make an exception if you can get a very low discounted fare which saves you at least 30 Euros per ticket, and you are willing to run the forfeiture risk).
www.trenitalia.com
There are also direct airport BUSES from FCO to both Naples and Florence. Buses to Naples run every two hours and drop you next to the train station, where you can continue to points further south by train.
www.fiumicinoexpress.com
There are also direct buses to Salerno, Battipaglia, Capaccio, Paestum and Agropoli:
http://www.expressbus.it/civitavecchia_agropoli.htm
BEWARE: From what I am able to discern, none of the bus stops in "Salerno" are useful for connections to the Amalfi Coast, as they are all remote highway exists suitable for people having relatives picking them up. If your destination is the Amalfi Coast, you should take the train to Salerno and continue on by boat or bus. If you are going to Vietri sul Mare, you could take the local metropolitana train from Napoli Centrale (operated by Trenitalia) direct to Vietri (thereby avoiding a connection in Salerno).
and direct buses to Assisi and Perugia:
www.sulga.it
and to more destinations ........
If none of the foregoing is useful, you need to connect at Roma Termini, Tiburtina, Trastevere, or Ostiense stations, in most cases, by taking one of the two airport train lines to those stations. Connections at Termini and Tiburtina are the most common.