Fast train tickets come with discounts - sometimes steep discounts - for early purchasing. Those tickets come with restrictions so you are basically trading flexibility for better pricing.
The two fast train companies are state run Trenitalia or privately run .italo.
https://www.trenitalia.com/en.html (use Italian city names Firenze for best results on Trenitalia)
https://www.italotreno.com/en
Both companies have apps that let you purchase and store tickets and track trains from your phone using Italian cell service or WiFi.
Passes will rarely save you money unless all your trips are last minute and you book higher class tickets. They make sense for businessmen or people who need to move around without much warning or want to book the highest level of service.
Price out the passes and see what works for you but for most people just buying tickets makes the most sense,
=Tod