As the others have indicated using advance purchase, discounted tickets is usually a far more cost effective method for travel in Italy. Rail passes are rarely the most cost effective method for travel in Italy. I suspect this situation is far different than your last trip to Italy 19 years ago.
However there are some potentially expensive "caveats" to be aware of when using trains and other public transit in Italy. For example.....
- tickets for the fast trains (Freccia, Intercity) have compulsory seat reservations, and are specific to train, date and departure time, and can only be used on the one train specified on the ticket. Therefore if you buy advance tickets on those trains, you must be willing to commit to a specific departure. This is also the reason why it's generally not a good idea to buy advance tickets to connect with an inbound flight. It the flight is late or you're delayed for any reason and miss your train, the tickets are worthless. If you board the wrong train by mistake, you'll be subject to hefty fines which will be collected on the spot!
- Regionale trains do not require reservations, but if you buy tickets locally, the tickets must be validated prior to boarding the train on the day of travel, or you'll again be subject to hefty fines. You can also purchase tickets for Regionale trains online, and these will come pre-validated. Tickets for Regionale trains are the same price whether purchased in advance or on the day of travel.
- also note that the cheapest advance tickets are non-refundable and non-changeable once purchased, so be sure of the details before you press the "buy" button.
- tickets for Metro, Buses, etc. must also be validated as these are also subject to fines.
For the routes you'll be travelling, you can also use the competing Italo Treno high speed trains. If you decide to buy advance tickets, you can either use the rail network websites or https://www.trainline.eu/ (they sell tickets for both Trenitalia and Italo, at the same price as charged by the rail networks).