Train Rome to Venice....

12 something is better than 11, but between the flight being a bit behind and lines at immigration, its probably best ( "safest") to buy the ticket there, even though it will not be discounted.

It will be more expensive but still reasonable. There are three levels of ticket prices -- Super Econ, Economy and Basic. There are so many variables that you cannot control that the best advice is buy your ticket on arrival. Personal I think this is little advantage to first class. It is a short ride. Second class is very similar in comfort to business class on a plane. If it is Oct 14, then most likely the two discount tickets classes will be gone so the Basic ticket would be the only one available so wait till she gets there.

For example the basic fare, 2nd class is 87E and first class is 120E. Fares also vary some by the time of day.

If you'll have a working smart phone when you land, install one of the train apps like Trenit (for Trenitalia trains) or Trainline.eu (for both Trenitalia and Italo trains). Then buy tickets on your phone and show them on the phone - no need to print tickets out. Or try to buy the train ticket for the whole trip at the airport train station. (Maybe not an Italo train from there though.)

For example the basic fare, 2nd class is 87E and first class is 120E. Fares also vary some by the time of day.

The walk-up fare for the direct FCO-Florence Trenitalia train departing at 11:08 is 56 E in 2nd and 77 E in 1st. One way, per person.

An image search like "frecciargento seats" will show why everybody says that 1st class is a waste of money for a 100 minutes journey .

As an alternative, they could take either the Leonardo Express to Roma Termini (14 E pp) or the FL1 local to Tiburtina (8 E pp). Then the first High Speed train departing to Florence, trenitalia or italotreno does not matter: whoever has a train departing first or whoever is cheapest. The price would be different, but not more than 62 € pp n the lowest class, airport train included.

Purchase your ticket after you arrive at FCO. Costs a little more, but if your flight is delayed or cancelled you will not be out the ticket price.

Second Class on the Rome to Venice train is more comfortable than coach on an aircraft. Free WiFi, comfortable seats, stroll down to the bar and have a drink while the train clips along at about 200 km/hour, steady as a rock.
First Class is more spacious, maybe a bit more peaceful, fewer children - you get what you pay for, I suppose.

Better to buy your tickets in Rome, trains run about every half hour or so.

You can go without booking.
If Florence is in your plans, I’d go to Florence first because it’s a shorter train ride and I wouldn’t book ahead because it’s hard to predict your arrival time by plane.
You can do Venice after Florence. In that instance, since you are more able to control your time by then, you can prepurchase tickets from there on.

Someone mentioned getting the Trenitalia app, and that’s good advice.
An app that I use all the time in Venice (and I’ve been here for a couple of weeks now) is Che Bateo (Which Boat).
You can select English in the settings.
Plug in your departure vap stop and your arrival stop, and you will be presented with Vaporetto (water bus) options.
The app always selects the fastest route, and it is accurate to a minute or two.

The new Trenitalia app is buggy and kind atrocious in some functions. Trainline app, which sells all regular Trenitalia and Italo tickets (with the advantage of comparing them when you search), works more smoothly. Just my 2c.

Thank you for all your good information, which I've passed on. She mentioned quieter, so she'll likely do first class. And I've mentioned trainline.eu for easy ticketing and she's hoping the 11:50 train will work.
Thanks again!