Storing Luggage

Trains never have luggage check-in. Everything is hand luggage. Like on the subway.
You have to carry your own luggage from the entrance of the station, to the platform, onto the train and stow it yourself. For more info see here: https://www.seat61.com/luggage-on-european-trains.htm
Some larger stations have luggage trolleys, but they are not common.

Very occasionally there is a luggage service between big stations, but it does not travel on the same train as you (the trains don't stop long enough to do that), instead you have to deliver it to the station ahead of time and/or collect it the following day.

Larger stations have luggage storage. You need to check each station individually.

If you are asking these questions you are under the misconception that trains are like planes. They are not. There is no check-in. You just walk in off the street and onto your train. You will not be stopped (there may be a ticket barrier in larger stations). It is your responsibility to find the correct train and get on. Nobody will stop you getting on the wrong train. You ticket will first be checked by the conductor some time after the train has left the station.

You must get your own suitcases onto the train, onto a luggage rack, and off the train. A 25" bag is far too large to go into an over-seat rack, so you'll be looking for space on the floor-mounted racks. I don't remember whether those on Italian trains are always near the doors; I've definitely been on trains with mid-car racks, which means you're wrestling the bag down a narrow aisle. You have to go up 2 or 3 rather narrow steps to get onto the train, and negotiating those steps isn't particularly easy even with a bag much smaller than you are proposing to use.

As you have guessed, I think two large bags are not the best idea. I'd want something smaller, and if that meant 3 or 4 bags instead of 2, so be it. Your children would be able to pull small wheeled bags.

I don't believe Italian train stations currently have storage lockers, but some have staffed left-luggage counters. You need to allow for a bit of time when both dropping off and picking up, and always look for a sign about hours of operation. I think you might encounter some mid-day closings.

If you know in which cities you'll need to store your bags, ask here for specific information about those. The situation does vary from place to place.

One thing to help your sanity is that many of the multi-track stations have elevators so you can avoid the stairs. With two kids this is a blessing. As others have said, two 25" bags are unwieldy and overkill. I’d get 21-22" bags and smaller ones or back packs for the kids. If the kids get tired of rolling their own (LOL) they should be light enough that an adult could easily carry one or stack it on their bag.

What you have to do is make sure you don’t pack anything that is not needed or heavy. Lay each family member's stuff out and then pare it down. If you forget something, you can always buy it in Italy. Larger stations normally have luggage storage but most often if you arrive before check in time, your hotel would store it for you at no charge.

Thank you so much for the link to the train luggage info! And the tips about 4 small suitcases rather than 2 large ones! Much appreciated!

Florence and Rome train stations have bag storage and escalators where necessary, but you'll probably go right to your hotel, anyway, and can store luggage at the hotel before/after occupying the room. The small towns along Lake Como (such as Varenna) and Cinque Terre stations don't generally have bag storage or elevators, but there's a flat track crossing at Varenna-Esino station.