Schedules

No need to buy any euros upon arrival, go to a bank ATM and get them there. If you want to feel better about not having to find bancomat (ATM) quickly, get about 50 euro from home just to carry you until you get to the ATM.

Many people buy all of their train tickets ahead of time for every trip EXCEPT their arrival day. There are too many factors to take into consideration you don't want to give yourself too little or too much time. Its WAY easier to just show up when you show up at the station at the airport and buy your ENTIRE trip from there, no need to wait until Termini to buy a ticket, especially since if you were doing it at the machines, they are farther away from the tracks where you would change.

Hello caalcot, and welcome to the boards!

Katherine's advice is spot on for these issues. Use your debit card to withdraw in euros. ATMs - called Bancomat in Italy - are everywhere but use those physically attached to a bank to avoid independent ATM fees whenever possible. Buying euros in the US is very expensive but may be worth the piece of mind. (I used to do this but credit cards are so easy now I don't even bother.)

I wouldn't book my "day of" travel from home because either you buy a low cost ticket which is non-refundable and you may have to just give it up if your plane is late or you book much later in the day to be safe and end up sitting around in Roma Termini waiting.

For high speed trains between large cities there are fast trains that come with specific times and reserved seats. These are either run by Trenitalia (Italy's train system) or the private company .Italo. Both are fast, have different classes, provide WiFi and have tickets that are cheaper the earlier you buy them.
https://www.trenitalia.com/en.html
https://www.italotreno.it/en
So it may be worth it to book everything ahead of time if you know your travel days between big cities. You trade time of day flexibility for saving money.

The slower local trains are generally called regionale and are booked whenever you want to use them. The prices don't change so don't worry about buying ahead. Trenitalia has an app that allows you to check schedules, buy tickets, and track trains in real time and I recommend it for convenience.

Hope that helps, have a great trip!
=Tod

....have no idea of average time getting thru immigration......, That is the problem. Generally you can get through in less than an hour BUT a couple of years ago, on a Saturday morning, it took over three hours. Therefore, you do not buy tickets in advance -- too much uncontrollable risk. You can buy your ticket at the train station ticket office in the airport. For the most part you will take the L Express to Termini and change to a train to Florence. I would allow at least an hour. Give yourself time to find the next train, use the water closet, and maybe a snack for the train. You can get coffee on board the train. There are a couple trains a day going to Florence from the airport without changing trains in Termini. You might get lucky. The train ride will put you to sleep.