From Rome FCO hire a private driver from the airport to Rome Termini station. About 30 minutes. Then Train from Roma Termini-Napoli Centrale. 1 hour 10 minutes. On arrival in Naples, a private driver to the Amalfi Coast would make the most sense. Around 1 hr 20 min drive. Use this site for train timetables.
https://www.trenitalia.com/en.html
Another option is train, Rome-Salerno 1hr33min
Then ferry to town of choice 35-70 min.
Bus and ferry timetables.
https://www.positano.com
There's a schedule change in December so you just need to choose the same day of the week you'll be traveling to see departure times and prices. This ticket needs to be purchased after you reach the station. If you pre purchase, and your flight is late the ticket is worthless.
Do you have an idea of what you want to see for this portion of your trip? Naples, Pompeii, places on the Amalfi Coast, Caserta Palace, Capri?
All the villages on the coast are small, there is good bus and ferry service to all of them. What type of lodging do you require? Individual hotel rooms, apartments or a villa.
If you can narrow it down, we can be of more help.