In Rome, the train from the airport is a "recommended" way to get downtown, however, you may find that for two persons, a taxi is a preferrable way to go.  When you consider the train fares, schedule, taxi  fare from Terminus, and the train hassles, you may want to consider hailing a cab.  Cabs are less stressful after a long transoceanic flight; they are reasonable and comfortable.

The train is about 11 EUR per person.  A taxi from Fiumicino is a 40 EUR  and 30 EUR from Ciampino, this is a set fare day or night - don't let them charge you more (they will try). 

Once you arrive in town, you can decide how to get to the attractions.  You should find Rome easy to walk EVERYWHERE.  You get to see more, meander in and out of shops, and it is no more than 45-50 minutes to any attraction you will want to see in major downtown areas.  You do need good shoes as the brick or cobblestone streets are tough on your feet and legs.