Train from Rome airport to Roma Termini train station

Yes this is very easy
You just take the Leonardo Express which will get you there fastest cost is 14 euros
or the Regional Train FL1 (requires a transfer to line FL5) cost is 8 euros

There is also a bus service that goes there:
http://www.terravision.eu/airport_transfer/bus-fiumicino-airport-rome/

thank you!

Or you can take a shuttle bus (SIT, TAM, Terravision or Atral) for 4-6 euro:

http://www.adr.it/en/web/aeroporti-di-roma-en-/pax-fco-bus

We've done it: very easy.

If, for any reason, you don't want to take the bus or the train, you can take a taxi. It's a fixed rate of €48, and no tipping is necessary beyond this. So, if you have more than two people, or a lot of luggage, or just are feeling tired and/or overwhelmed, take the taxi. Just make sure to get in the official taxi line, and do not accept any offers of rides from anyone else.

The Regionale F1 does not go to Termini, but Tiburtina and a few other Rome train stations.