Hello wesavelives, and welcome to the forum!
Fast trains between big cities and these come with assigned seats on a specific train and tickets can be purchased early at a (sometimes sizable) discount but come with restrictions. The earlier you buy the cheaper the tickets but you're trading economy for flexibility. Tickets for fast trains are generally available two months ahead.
Local trains called regionale tickets are always the same price - just buy them when you need them. Also if you have a reserved train and you miss it because of the train being late and it's the train companies fault they are required to re-book your ticket even if it is a restricted change ticket. Of course if you just miss a train with a restricted ticket the restrictions apply.
You'll want to use the Italian city names for best results when searching on train sites: Roma for Rome, Firenze for Florence etc.
Get used to using 24 hour "military" time when dealing with trains but it's used less in Italy outside of trains.
The man in seat 61 is a really good basic primer on European train travel: https://www.seat61.com/train-travel-in-italy.htm
Have a great trip,
=Tod