Italian Train Ticket Details (Free Stopovers?)

[KruzenKaren]

Hello, I plan to take the train in Livorno and Naples and I'm a little confused about buying the tickets. I'm not an experienced bus/train rider. I have pulled the schedules on trenitalia.com
1.In Livorno I plan to take the train first to Pisa, then to Florence. After touring Florence I will take the train back to Livorno. Do I buy a train ticket for a round trip Livorno to Florence and it's free to jump off for a bit in Pisa? Or do you buy a ticket for each segment ie( 1 ticket for Livorno to Pisa, and another for Pisa to Florence, the one last ticket from Florence to Livorno = 3 tickets).
2. In Naples I plan to take the train to Pompeii and then jump back on to Sorrento. I don't need a ticket back because I'll be taking a hydrofoil from Capri. (Again do I need 1 ticket for Pompeii to Naples and then another for Naples to Capri = 2 tickets).

Bottom line are stopovers free? Do you just stick the ticket in the machine to revalidate each segment?
Thanks so much in advance cruisers for your expert advice!

[ftroute]

The best way (IMHO) is to buy tickets for each segment. Does not cost any more or any less. A single ticket can eliminate any confusion with a conductor (strange, but true.) Tickets on the Regional trains are good on any regional train (of the same class.)