Train is the fastest and easiest way to get from Rome to Fiumcino. Bus is also an option, but takes twice as long. I've included detailed info on getting to the airport from Rick's Italy guidebook below:
Getting Between Fiumicino Airport and Downtown Rome
By Train: The slick, direct Leonardo Express train connects the airport and Rome’s central Termini train station in 30 minutes for €14. Trains run twice hourly in both directions from roughly 6:00 to 23:00.
Going from Termini train station to the airport, trains depart at about :05 and :35 past each hour, usually from track 23 or 24. Check the departure boards for “Fiumicino Aeroporto” and confirm with an official or a local on the platform that the train is indeed going to the airport (€14, buy ticket from any tobacco shop or a newsstand in the station, or at the self-service machines, Termini-Fiumicino trains run 5:35-22:35). Read your ticket: If it requires validation, stamp it in the green or yellow machines near the platform before boarding. From the train station at the airport, you can access most of the terminals. American air carriers flying direct to the US usually depart from T5. If you arrive by train, catch the T5 shuttle bus (navetta) on the sidewalk in front of T3--it’s too far to walk with luggage.
Allow lots of time going in either direction; there’s a fair amount of transportation involved (e.g., getting from your hotel to Termini, the ride to the airport, going from the airport train station to check-in, etc.). Flying to the US involves an extra level of security--plan on getting to the airport even earlier than normal (flying transatlantic, I like to arrive 2.5 hours ahead of my flight; within Europe, 2 hours is usually sufficient).
By Bus: Various bus companies, including Terravision (www.terravision.eu), SIT (www.sitbusshuttle.com), and Atral (www.atral-lazio.com) connect Fiumicino and Termini train station, departing roughly every 40 minutes. While cheaper than the train (about €5 one-way), buses take twice as long (about an hour, depending on traffic) and can potentially fill up (allow plenty of extra time; leaves the airport from T3, leaves Termini from station’s north side). The Terravision bus also stops near the Vatican. At the airport, the companies' desks line up in T3, near the entrance to the train station; I'd just compare my options and hop on whichever one is departing first.
Once you are at the airport, I would say that taxi would be a good option for getting to your hotel near the airport. You could also check directly with the hotel to see if they offer a shuttle service that might be able to pick you up.