We love Trenitalia, the Italian train system !
We use it all the time when we are in Italy, both to transit between major cities like Florence and Rome, as well as for day trips out to smaller towns.
Trenitalia has a very complete website with 24 hour a day schedules where you can choose your station of departure
and destination, and print out the schedules at home.
You can purchase tickets at the stazione, or as we do, purchase online and print out the tickets at home before you leave home,
avoiding the lines at the ticket window, or negotiating the self-service kiosks in Itailan !

This image shows the Frezziarossa, one of the high-speed "luxury" trains we prefer, that make minimal stops between major cities.
We use "first class" where we reserve our seats and purchase our tickets from our home computer.
A snack cart comes by with cookies and light drinks, and there is also a car selling sandwiches, salads and light meals.

Smaller "one class" trains , service smaller towns, making these smaller places accessible to most anywhere you want to visit.