do I buy a train ticket from Rome to Ferrara after I arrive at the airport or before?

You will make that train or you won’t. The only thing buying it ahead of time gets you is some savings on the price, but cheap fares are not changeable.
Fast trains rarely sell out so you can either buy a cheap fare and eat the ticket if you miss it or buy a full, changeable fare which is the price you will pay on the day anyway.
If you buy a full fare and miss your train you can change your ticket for a later train. I’ve only had to change a ticket once in Milan but I have to say the ticket office in Milan was like a Fellini movie.

I understand the temptation but I would just wait. Get the Trenitalia app for your phone and buy the ticket on the train from the airport or once you’re actually at Rome Termini.

Good luck,
Tod

The Trainline app is ok for comparing since it aggregates information across platforms, but it's always best to purchase directly from Trenitalia or Italo. Happy travels!

Just to clarify.

If you buy a full fare BASE ticket, yes you can change it as many times as you like BEFORE THE TRAIN HAS DEPARTED, and you can do so also on the app.

You can also change it WITHIN ONE HOUR AFTER THE TRAIN HAS DEPARTED but ONLY AT TICKET OFFICE AT THE STATION (not on the app or self service machine).

So, if your flight is so badly delayed that you arrive over one hour after the train has departed, your money is still lost.

Since you are arriving at the airport, Italo does not serve it, only Trenitalia. Therefore, if you want a single ticket, only Trenitalia will do. If you want to use Italo, you need 2 tickets, one from Trenitalia to Roma Termini, then a ticket from Termini to Ferrara. Italo trains are less frequent than Trenitalia trains, so best stick with Trenitalia, even if it might cost a couple extra EUR.

I’d just buy it on arrival. Load the trenitalia app on your phone so you can check the schedule and buy the ticket on arrival.

At least half a dozen times we have flown into Rome or Milan, taken the train to Termini or Milano Centrale, then bought train tickets to wherever we were headed. No problem.

Once we pre-bought the really cheap early train ticket from Milan to Venice, figuring that if we missed the train we weren't losing much money, but we easily caught that train.

Once we flew into Rome, bought a ticket at Termini to Civitavecchia, waited on the platform listening to announcements that the train was coming just delayed, then our train and subsequent trains were canceled due to a strike. So, if we had bought that ticket ahead of time, we couldn't have used it, but we couldn't use the ticket we bought at the station, either. We ended up taking a taxi.

Last October, we flew to Rome, trained to Termini intending to buy tickets to Lecce in Puglia, discovered that the only train that would get us there in time was 100% full. Very unusual for a train to be full. We had to rent a car.

So, stuff happens. We've had two annoying plane-to-train experiences, with no clear moral as to whether it's good to buy a train ticket in advance. If it makes you feel less nervous and if you can afford to lose the price of the ticket and if you give yourself maybe 3 or 3.5 hours from scheduled touchdown to scheduled train departure when booking the ticket, I say go for it. Look into what trains leave for Ferrara later than the one you really want, just in case. Termini's Mercato Centrale is a great place to wait for a train and eat a snack or a meal, and to liken it in any way to a mall food court in America is to malign it.

Actually, the Trenitalia Economy ticket does allow changes to the departure time, but you must pay any difference in fare. So if the flight is significantly delayed, you can change the ticket from the Trenitalia app on your phone - you'd pay the same as if you waited to purchase on arrival. And any change must be on the same category of train. So use the Trenitalia app to book your ticket rather than Trainline.

"Changes to booking/ticket:up to the departure of your train, changes may only be made to the date or time of departure for the same category of train or service purchased. In this case, you are required to change your ticket (at sales channels where this operation is permitted), by paying the price difference between the original ticket price and the available Base price for your selected change. The new ticket will have the same characteristics as the Base ticket."

Edited to add: if you're looking at the Intercity train from Termini to avoid another change, note that the journey is two hours longer than if you booked an itinerary using the high-speed train to Florence for the first leg. And you couldn't change your Intercity Economy fare to a Frecce fare. Changing trains in Florence is easy, so factor that into your planning.

Thank you everyone for sharing your experiences and information. I am much better informed.