Hello dejackome,
There are fast trains for all of these routes. There are two fast train companies in Italy - the state run Trenitalia https://www.trenitalia.com/en.html and the private .italo https://www.italotreno.it/en
Most people agree is no good train pass for Italy anymore that is worth worrying about. You can simply buy tickets for each leg easily and, if you buy early, cheaply. Fast trains have classes that come with better amenities but all tickets come with assigned seats for the duration of your trip. None of your trips should be longer than 2 hours so I wouldn't worry about "upgraded seats" in different classes if you're comparing costs.
The earlier you buy you tickets the cheaper they will be so avoid buying "day of" if you can. Because they have assigned seats it is possible for these trains to sell out but that would probably only be on the busiest days of the year. Check the details of the different tickets - the cheaper the tickets the more restrictions about changing them later so you're trading price for flexibility.
It's really pretty straightforward just check the schedules.
Remember to search using the Italian names - Firenze for Florence, Roma for Rome and Venezia for Venice on Italian sites.
Trenitalia has an app for your phone and I'd be surprised if .Italo didn't have one as well.
Have a great trip,
=Tod