Also should mention that if you purchase tickets on the website without an account - or through a reseller like Trainline - you can transfer them to the app on the phone using the ticket information. If you use your account on the Trenitalia site they should link and show up in your phone automagcially.
Keep in mind that if one person is the holder of the tickets they will need to scroll through the various tickets on their phone for the conductor. It's not a big deal and conductors are used to this but make sure it's the responsible person who will have their phone and it is charged etc. You can also usually transfer the tickets to the various wallet apps on your phone as well.
Just to clarify about the seating - if you are in second class headed to Bologna on a Sunday the car will probably be filled with college kids visiting, catching up or trying to study and no one will care about seats. If you're riding first class to Venice it will be primarily tourists and everything will be orderly and vastly over luggaged. It's hard to describe but certain trips have different vibes. It's usually orderly and organized and then occasionally explosively chaotic. Like the train that had different cars than expected attached to it and other people had the same seat assignments we had and the only arbitrator was a harried employee with a clipboard with unassigned seats on it.
We're probably vastly over-preparing you for a simple 2 hour trip from Milan to Florence that will be uneventful and straightforward, but navigating the occasional chaos of the Italian train system is sometimes part of the adventure. Riding to Florence sitting together will probably help pass the time but riding back may be quieter after all day in Florence.
Prebook tickets for any must see sites in Florence so you don't miss anything or waste time in ticket lines, have a great trip,
=Tod