Train info for Rome (Termini Station) to Verona Oct (Sat) & Verona to Padua to Venice Oct (Sun)

Very simple. Go to the the TrenItalia website. To see trains from FCO, enter Fiumicino Aeroporto or Roma Termini for departure and Verona Porta Nuova for destination. Depending on which train you chose, travel time can be 4 to 5 hours with 1 or 2 changes. On your next leg, enter Verona Porta Nuova as departure (Regionale trains run twice an hour and travel time is about 1 hr. fare is €7.10), Padova as destination, then Padova to Venezia S. Lucia (3 or 4 Regionale trains per hour, 30 minutes travel time, fare is €4.15)

There's no need to buy 1st class tickets as 2nd class is quite comfortable and significantly less expensive. The sooner you purchase tickets, the cheaper they are. On the trip from Rome, you can buy 2nd class seats as cheap as €49.90 Super Economy. You would have to commit to a particular train and changes are not permitted. Realize that you're at the mercy of airline travel and your flight can be delayed and immigration/customs at the airport can take a while. If you miss the train, your ticket is worthless.

You can buy your tickets to Padova and Venice at the station on the day you travel. Those trains are Regionale and neither have reserved seats or require reservations.

Consult the website of the Italian railways (Trenitalia) for schedules and prices. (You must use the Italian originals for the station names: Roma Termini, Verona Porta Nova, Padova and Venezia S. Lucia.)
There are direct trains from Termini to Verona, taking some 3 hrs. You can buy those tickets in advance with Trenitalia at huge discount, but those will be non-refundable tickets only valid for that specific train (day and hour). Assuming you fly in to Rome, you can't be sure at what time you'll be at Termini: late arrival, long line at Immigration, or whatever. The so-called Base tickets are much more expensive, but allow you to change your tickets for a later train.
Verona to Padua and Padua to Venice is best done by Regionale trains. The distances being so small, they take a few minutes more than the much more expensive express trains.

When you look at the schedules at Trenitalia for Verona Porta Nuova to Padova, and Padova to Venezia S. Lucia, you will see that some of the trains are marked Regionale Veloce. Plan on taking these as they make limited stops and are almost as fast as the Freccia trains. The ones just marked as Regionale are a lot slower because they make every single stop along the way. Price is the same.

We want to go from Rome to Verona on day 1 Day 2 Verona to Padua
Padua to Venice.

Marla, I'm curious why you want to do this? Without knowing what time your plane is scheduled to land in Rome, you may end up with little time in Verona, and what you have of that being a jet-lagged haze.

Verona to Padua to Venice will mean parking your luggage somewhere in Padua for sightseeing: have you explored whether that's an option (could be but I'm not certain).

It's your trip so you should do want you wish to but I wonder if there's some wisdom in just staying in Rome on arrival day - to catch your breath - or going directly to Venice, which is probably what I'd do. You'd get an extra full day in Venice, and there's enough to explore there outside of what the tour covers to make it worthwhile.

sorry for any confusion. We are spending the night in Rome after the flight & leaving for Verona the next day.
Thanks to all for the tips!

you will see that some of the trains are marked Regionale Veloce. Plan on taking these as they make limited stops and are almost as fast as the Freccia trains

Yes, I suppose, but the Freccia trains are much nicer. And if you can buy in advance and commit to a schedule, you can often get a significant discount.

sorry for any confusion. We are spending the night in Rome after the
flight & leaving for Verona the next day.

Ah, thanks for clearing that up! I'm never sure if someone is counting their "Day 1" as arrival day or first FULL day on the ground. :O)