using trains in Italy

Well, in that example, Lucca is not on the way from Rome to Florence. It would be an easy day trip from Florence, though (some would combine it with a day trip to Pisa). You can't hop off and on the Frecciarossa trains with reserved seating - you'd need to buy separate tickets if you stop somewhere. With a regional train (where you are subject to random inspections, no reserved seats), you MIGHT get away with hopping off and back on if your ticket wasn't scanned on the first train and you board a train that left your original station less than four hours after the original train. But I would expect to need to buy a separate ticket there too.

Many train stations have left luggage counters or lockers where you can leave bags. If one particular train station does not offer that, you might find alternative, safe places to leave luggage. I left my bags at a bar once in Germany in a small town while doing a hike for several hours - it was the only place open in town on a Sunday. They refused to take any money for it when I returned for my bags, so I bought something to go.

You would not pass through Lucca if you were taking a train from Rome to Florence. The frequent fast Freccia trains are basically nonstop to Florence after you leave Rome Tiburtina station. So check into your hotel in Florence, and if you want to go to Lucca, go back to the station and buy a ticket. There is a direct train every hour at 10 past the hour and takes 1 hour and 19 minutes. Slower trains at other times.

No lockers at train stations in Italy, as far as I know.

Lucca/Pisa is an easy train or bus trip from Firenze, before you set off for Roma.

Ideally Lucca is best done as a stand alone day from Florence.
If your travel day from Rome to Florence is the only day you have to get to Lucca then get an early start. Store your bags at Florence SMN and then head on to Lucca by local train. Pick up your bags late in the day on your return when proceeding to your hotel. The baggage store at SMN is easy to find and staffed til 23:00.
http://www.grandistazioni.it/cms/v/index.jsp?vgnextoid=86470fc45fa4b110VgnVCM1000003f16f90aRCRD

A cut price high speed service = http://www.italotreno.it/en also runs on the same tracks by a company that competes with the national network = Trenitalia.

I wouldn't call Italotreno "cut-price". Their prices are in line with Trenitalia's. But some do prefer them.

Regional train tickets are very cheap and are priced on distance so buying a ticket A to C is nearly the same prices as a ticket A to B and B to C. Don't try to play games as suggested above. If caught the fine is substantial and payable on the spot.

If you want to look around Lucca for a few hours, it's easy to do a day trip from Florence. That way you can leave your luggage in your hotel room.

One other point to mention is that if you buy tickets locally either for Regionale trains or Buses, you must validate these prior to boarding on the day of travel. Unvalidated tickets are subject to a hefty fine, which is collected on the spot.

There is no need to "play games" or to "get away".
A Regionale ticket locally purchased to go from A to B must be time-stamped just before getting on.

From that moment the ticket can be used on any regionale going from A to B and departing within the next four hours.

Stopping at an intermediate station would be pefectly legal, of course you must get on another Regionale going to B within 4 hours.

Since the bomb in Bologna there are no more lockers in Italian stations, only staffed deposits in the main ones.