Rome Airport website is www.adr.it

Shuttle from Fiumicino station to Rome Termini train station is called LEONARDO EXPRESS. It runs every 30 minutes (from 6.37 to 23.37). Cost is 11 Euros and travel time is 31 minutes (no stop).
Do not take the "metropolitan train" as this does not go to Termini station !

From Roma Termini Station, you have 2/3 trains every hour to Firenze SMN (Santa Maria Novella).
ES (Eurostar) trains are a bit more expensive but faster (1.40 minutes), wheras IC (Intercity) trains take 2.30 minutes.
Do not take Regional or Interregional trains as they stop at every station !

In Italy the difference between 1st and 2nd class is on the size of the seats and compartments. On ES trains, you will probably get free newspapers, snacks etc.
1st class tickets cost double than 2nd class.
In general, ES and IC trains are OK. Service, cleanliness, punctuality are bad on regional trains.
It is true that italian railways are not as good as in France, but they are also much cheaper !

Once you are in Sardinia, you can buy any railways ticket from a local authorised travel agency (check sign FS).