Just do yourself a favor and price it out beforehand. A single-country eurail flex-pass will almost certainly be cheaper than individual tickets if you want to use trains extensively:
https://www.eurail.com/en/eurail-passes/one-country-pass/italy

Basically, the way these passes work is that you have a limited number of "travel days" where you can do all the train trips you want. And you have to use those up within a month. If you are there for several months, you can choose the number of days per month you want to use.

For me, it was easy enough to board with the pass without booking in advance, but I wasn't traveling in peak season and I only had one or two travelers. I think you'll want to book reservations in advance for peak travel season, even with the eurail pass.

Eurail is for non-eu residents. Interrail is what the service is called for EU residents, and I believe that is somewhat cheaper.

The more people you have, the more a car rental/multiple short car rentals starts to make sense. However, keep fuel prices in mind; if you're used to US fuel prices, it's not gonna be half that cheap.

And for long-distance travel, train is obviously superior from an experience standpoint.