Hi eljohtheron -
"ItaliaRail" isn't a train system; it's merely a third-party booking agency, and one a fair amount of us on this forum do not use for purchasing our tickets. For your journeys, you will be using Italy's national rail system: Trenitalia. For fast trains between bigger cities, Italo trains are also an option, and there are some smaller localized lines as well, although you won't be needing those.
But yes, there is room for luggage on any carriage of any Trenitalia train, or any train at all in Italy. You may have to store your large bag in a space for those at the front or back of the carriage, if it doesn't fit above your seat or, when provided, spaces between the seats. Keep your hand luggage with you. On regionale trains (no seat reservations) it's possible that you could stand for awhile if the train is very full, although we've yet to run into that situation.
We always travel 2nd class on the high-speed Italian trains as the space difference isn't enough to warrant 1st class or business prices.
Editing to add: regarding those "large" suitcases, do be aware that the larger and heavier they are, the more difficult they're going to be to manage in Italy. Many hotels do not have lifts, and you'll need to be able to get them in and out of the trains pretty quickly. Stairs are going to be unavoidable, here and there, and walking surfaces can be rough (e.g. cobblestones), narrow and/or uneven. Pack as lightly as possible, and each of you should be able to manage your baggage yourself.