1. To get to the train into Rome from Fiumicino Airport: After passing through customs one enters a large hall. Turn right  and find stairs  and escalator to an underpass at the left. Go through the underpass and on exiting there is a desk which sells train tickets.

2. There is a convenient Internet Point near (or maybe part of) the Hotel Aphrodite on Via Marsala which runs along one side of the Termini Train station  in Rome.

3. You can buy the romapass and get free admission to 2 sites and discounts to others plus a 3 day transportation pass.  Check out romapass.it