What is the best way to get from Fiumicino airport to a hotel in the Pantheon neighborhood by Palazzo Farnese.
Linda you could take the train, make sure its "Centeral Roma Termini" for 14 euro. I would get directions from the train station to your hotel, maybe a cab from there. Sometimes the hotel will send a pickup from the airport, you may want to check the rates or if you are a group check www.airportshuttle.it 35 euro first person and 6 euro each extra, direct to the hotel.
You have several but only you can determine best. For 14E the Leonard Express (train) will take you to directly to Termini - no stops. Follow the signs to the train station next to the terminal. Also, a local train leaves from that station making a couple stops for 5.50E but it also ends at Termini so you don't worry about getting off. Two problems with the train - fairly long walk through the train station to the front and then a taxi to your hotel. Next option would be a taxi to your hotel. It is a FIXED rate of 45E for one to four passengers. With four passengers you need to be light on luggage. Another option is the SITBUS to Termini for 8E. Leaves in front of the airport. But still need a taxi to your hotel I am not aware of any hotel who will send a car to pick you up for free. Your hotel can arrange a pickup with a private car and driver. You can also arrange a private car and driver. Romecab is often recommended. We always use the train but we we also stay in the Termni so we can easily walk to our hotel.