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You don't have to go all the way to Civitavecchia and then return to Rome. That's eating up a lot of your precious time, and there's no direct train from FCO to Citivecchia -- you have to go through Rome. The trip will take you around 90 minutes each way, at best.Thanks! That sounds like the thing to do to save time and expense!
I suggest you take either the train or one of the express bus services (cheaper and with luggage storage space below) from the airport to Termini station in Rome. There you can check your luggage while you sightsee in Rome for the day.
When you're done with your sightseeing in Rome, go back to Termini, claim your luggage, and get on the train to Civitavecchia.
Here is some additional information on the luggage storage at Termini: