Very easy to do. The B.I.R.G. is the Regional trains ("R" on the schedule on trenitalia's site) and includes buses/metro in Rome. A return ticket from Civitavecchia to the main station (Termini) in Rome, including use of buses/subways in Rome is €9. Look up train times on trenitalia website and get a good street map and off you go.
We never done that but read a few reviews people get train too rome for few ueros but allways got to be back on tme to ship you ould allways seif you can chm up with some one else then go your own way then meet back up with them to travel back we done that one year to see the tower of pisa it works out quite cheap there was about ten of us.
Mr young from leven go on reviews quite lot of info about rome doing trip on your ow.
Be aware that queueing to get into the Vatican museums (as opposed to St. Peter's Basilica) can eat into your time.