rail pass

Thanks for the info.....It seems the best value would be to wait until we get to the train station, if I'm understanding you correctly. There will be one senior and one adult traveling. We will be going directly from Rome's airport to Sorrento, changing trains in Naples.

Definitely don't do a rail pass. Buy point-to-point tickets. (That's a more accurate description than "pay as you go," for reasons described below.)

As for the individual tickets, you can save money on the high speed rail lines by buying discounted but nonrefundable tickets that become available 4 months in advance of the travel date. For that, use trenitalia.com.

You can also wait until you arrive in Italy and buy all your train tickets at the same time, paying full fare. Buying them all at the same time is safer than waiting until the day of travel to buy them, because you never know how long or slow the ticket line is going to be that particular day.

As for your arrival date, do you understand that you can't take a direct train from the Rome airport (FCO) to Naples? You'll have to train into the Termini station in Rome, then change to a train to Naples.

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I few additional points to mention....

• For the trip from Rome to Sorrento, you'll first use the Leonardo Express from FCO to Roma Termini, where you'll connect with a high speed train to Napoli Centrale. You can buy tickets for both trains at the airport rail station. Be sure to validate your Leonardo Express ticket prior to boarding or you'll face hefty fines which will be collected on the spot!
• For the trip from Napoli Centrale you'll need to go downstairs to the Garibaldi station and buy tickets on the Circumvesuviana, which is a commuter train. You'll have to buy separate tickets for that leg of the trip. The Circumvesuviana does NOT accept Rail passes. Be sure to watch your luggage on the Circumvesuviana, as there will likely be pickpockets and scammers about.
• For the trip from Sorrento to the Cinque Terre you'll have to reverse your route to Roma Termini. There are a few direct, high speed trains every day from Roma Termini to La Spezia Centrale, with a travel time of about 3H:45M. I'd recommend using one of those. You can see the schedules on the Trenitalia website. From La Spezia it's a short ride to the five towns. Which town are you staying in?
• For the trip from the C.T. to Florence, you'll likely be using about three trains. Note that your destination will be Firenze S.M. Novella (many cities have more than one station so you need to know which station you're using when you buy tickets). Some trains on that route may be Regionale and some may be "premium" trains such as Freccia or InterCity which have compulsory reservations. Again, with the Regionale trains you MUST validate your tickets prior to boarding on the day of travel or risk hefty fines. With the premium trains, the reservations are specific to train, date and departure time so you can ONLY use the one train listed on your ticket. If you board any other train, again hefty fines!
• For the trip to Cortona, you'll likely be travelling on a Regionale (don't forget to validate!). There are two stations to access Cortona, Camucia or Terontola (I prefer Camucia). Once you arrive at the station, you'll need to take the Shuttle up the hill, and buying bus tickets can sometimes be a bit of a challenge. Going back to the station isn't a problem as there's a Tabachi just down the hill from the Piazza where the Buses depart, and they sell tickets.
• For the trip back to Rome, there appears to be one direct departure at 10:32, arriving Roma Termini at 12:46 via a Regionale (don't forget to validate!). That's the one I'd use as many of the other trains have one change in Florence or elsewhere.

It's possible to save money by pre-purchasing tickets on the premium trains, but note that those will lock you into travelling on a specific train and time. Regionale tickets are best purchased locally using either a Kiosk or staffed ticket office, as there are no price breaks for pre-purchase.