Traveling by train in Italy means always traveling with carry-on luggage - you'll carry on board everything you have with you. It's your responsibility to bring your bags on board and stow them properly, which is a great incentive to pack light.
There's no fee to bring luggage on the train, nor do you need a special ticket to carry luggage on board. The limit on bags each passenger may carry on board is technically three pieces of luggage - but the limit is rarely if ever enforced. Still, it's a good idea to keep bags at a minimum to ease the loading and unloading process.
The on-board luggage storage options differ depending on the type of train. On the newer high-speed trains there are storage areas at one end of each train car, often across from the toilet, which are available for anyone in that car to use for their larger bags.
On the older trains, the only real storage areas are overhead shelves above the seats. Sometimes there is room to stow medium or small bags in the space created when two seats are situated back-to-back. Smaller bags—backpacks, purses, briefcases—can be slid underneath your seat.
Trenitalia isn't responsible for lost bags, so it's your responsibility to keep an eye on luggage even if it's stowed at the end of the train car. There is, however, insurance against bag theft that applies to most Trenitalia passengers. If a bag is taken from the storage compartment at the end of the car of a Frecce or InterCity train, the insurance maximum is €300 for a single bag and €600 per customer. The maximums for other trains are lower, and the rules for filing an insurance claim are strict.
There is a door-to-door option for sending your bags from point to point rather than carrying it with you. It's called Bagaglio Facile, and the costs start at €20 for the first bag and €15 for the second. It's available to all travelers on Frecce trains, or anyone who has a CartaFreccia card.
Bagaglio Facile is only available Monday-Friday (it only operates on weekends in the cities of Milan, Rome, Florence, Venice, Turin, Bologna, Naples, Salerno, and Bari). Arrangements for Bagaglio Facile must be made by phone at least 24 hours in advance of your departure time.
If you're stopping midway on your journey and want to do some sightseeing before your next train departs, many stations offer some kind of luggage storage service. Luggage lockers are no longer the norm for security reasons, but there are often offices with staff where you can leave your bags for a period of time for a fee.