Hi, Sara,
I have trained it all over northern and central Italy, over the last several years, either with another woman (sister, daughter, sis-in-law) or by myself (two weeks summer 2015). Here are some of the most useful things I've learned about trains in Italy. First, there is a big difference between the faster trains that require reservations, and the more local, "regionale," trains that do not take reservations. With the faster, (usually "frescia" on Trenitalia) trains, you must use the train for which you have the reservation - no changes allowed - and sit in your reserved seat. You do NOT have to validate that ticket. You just show it - or the reservation number - to the conductor when he/she comes through the train. The "regionale" tickets however are good for some number of days (weeks?) after purchase, and can be used on ANY regionale train that goes to your destination. Thus, if you are, say, going from Pisa through Florence to Siena, and you miss your planned connection in Florence, you can take any later regionale train on to Siena. When you START your journey, your regionale ticket MUST be validated in a green machine near the train platform, just before you go get on your train. (In a smaller station, these may be in a passageway underground, before you walk up the stairs to your platform. I'm thinking maybe Lucca?). You do not have to revalidate at your connection.
Now, here's the most useful note I can share: When you buy your ticket, quite likely from the kiosk in the train station (the Florence kiosks are great; you can load up on several week's worth of tickets, right there), the ticket that you receive ONLY has your departure and arrival stations on it, with ONLY departure and arrival times at those stations. It does NOT have your connection stations and times on it. SO: Have a piece of paper and pen in hand, as you are scrolling through train times at the kiosk, and WRITE DOWN the connection station, with ARRIVAL and DEPARTURE times at that station, and with the train number for your connecting train. These will NOT show up on the final screen before you hit "purchase," and will not be on any part of your ticket. (If you forget to do this, you can go back to a kiosk and start the process over again, to check where and when your connection is, etc. without going on to final purchase.) You may have a combination of "frescia" and "regionale" trains on one trip. Be sure to write down all connection info while you are purchasing the ticket, whether from a ticket agent, or from a kiosk.
Finally, 15" is usually plenty of time to make a connection, as long as the train is running on time. I have never yet missed a connection. In larger stations, once the train platform is shown on the big central computerized screen, then head quickly for your platform. It may be shown only 10" or less before departure.
