When you land you buy a train ticket in the airport for the De Vinci express (it's not fast) for E11. This will take you to the Termini. I was advised to buy two single tickets in the airport so as not to have to bother buying another one when I was going back (good advice). You have to validate one of the tickets in a yellow machine before boarding the train which pulls into platform 24 of the Termini. This platform on the extreme side of the Termini. Likewise, this is the same train you will catch back again on platform 24.
In the underground station you can but a ticket from the machine for E4.00 that allows you unlimited travel on the underground, bus and tram for the day. Beware gangs of pickpockets on the underground which will be very overcrowded. Keep your wallet out of your trouser pockets
Murt