Commuter trains run right into the middle (literally) of Fiumicino. You can take an express train that goes to the main train station, Termini - actually, they've renamed it Giovanni Paulo II station, I think. Or, for quite a bit less money, you can take a commuter train. The downside to that is that it makes a couple of stops on the way so it takes longer and it doesn't go to Termini. But if you are going to Trastevere, for example, it's fine.