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!

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.)