What train station in Rome is closest to the airport (FCO)? Tiburtina, Rome city or Termini
What is the easiest way to get from the airport to the train station going to Naples?

You'll take a 30 min train from FCO to Termini, and then your 2nd train will take a little over an hour to get to Naples.

A quick way to check your options to get anywhere is to check www.rome2rio.com

Taxi €48 if you choose that option. Tickets Roma Termini to Napoli Centrale, best purchased on a Trenitalia machine, as online tickets worthless if you miss the particular train you have booked.

Rome2rio is notoriously unreliable.

The FCO to Termini Leonardo Express will get you there in half an hour and costs 14 euro.

There is a 7 euro bus as well. Several companies do the run to Termini.

High speed rail gets you to Naples in just over an hour if you can afford the base fare - which is what you will pay as a walk up. The IC trains take two hours but are cheaper.

This site is a great information resource (but not for ticket purchases) for European train travel. Here's the specific page for Italy, showing some Naples info:

http://www.seat61.com/Italy-trains.htm#.V3JesWQrKRs

We are scheduled to arrive in Rome at 7:45am. How much time will it take to pick up luggage, get through customs and travel to Termini? I'm trying to determine what train we should aim for from Rome to Naples. We will also be traveling from Naples to Sorrento that day. I would like to arrange for a private driver ahead of time so I'll need to have a projected arrival time in Naples.
The other option his scheduling a private driver from Rome to Sorrento.