fordnegs, this is where you can explore train schedules:
http://www.trenitalia.com/tcom-en
Trenitalia is Italy's national rail system. There are some other small local systems - such as the Circumvesuviana in Campania - and one private line (Italo) that runs between the larger cities but this is the usual go-to site. You do have to use the Italian names for the cities, and the types of trains you'll use may or may not require seat/carriage reservations. For instance, your journey from Rome (Roma) to Florence (Firenze) on the fastest Frecciarossa or Frecciargento trains will but Florence to Lucca would be on a Regionale that will not.
Italo runs between Rome and Florence so you can explore pricing on that site as well:
http://www.italotreno.it/en
For the fastest, most expensive trains there can be a benefit in pre-booking if super-economy or economy rates are available but that does lock you into a specific day/time/train. 2nd class is just fine: not a big enough difference between that and 1st class to pay the price. Tickets for trips on regionales are inexpensive and should just be purchased the day of the journey.
Lots to know about Italian trains so don't be afraid to ask!