what is the easiest way to get from Sorrento to the Rome airport

By train, the route is Circumvesuviana from Sorrento to Naples/Napoli Centrale, on to Rome/Roma Termini. Transfer to Leonardo Express to FCO.

I suppose the absolute easiest would be a car & driver, direct to the airport, if cost is no object.

I should also mention, there are 2 airports, that I know of, for Rome. I believe the low-fare airlines use the one south of the city, Ciampino. There are trains to nearby Ciampino Citta station from Termini, and a shuttle bus to the airport. Otherwise, the route would be the same by train from Sorrento.

One should be very cautious about leaving Sorrento in the AM to make a flight out of Rome he same day - esp if transatlantic - when you typically have to arrive at the airport 3 hours in advance of the flight.

Quite true. The Circumvesviana from Sorrento to Naples is the commuter train and is very crowded in the morning. I suggest you do it mid-morning or later and on to Rome Termini, spend the night before your flight in one of many decent hotels walking distance from the station. And go from there on flight day.

This is one I like if you prefer a budget hotel. Easy walk to Termini, good neighborhood:
http://hotelgiuliana.com/default.aspx?lang=en