We will be flying into Rome Fiumicino early in the morning with another flight out of Ciampino late in the day. If we take the train to Rome Termini, is there anywhere that we can store our luggage until we head for Ciampino later in the day?
Yes, there is bag storage at Termini.
But plan for a long line in picking up the luggage. We have never used it but prior postings have suggested long lines for both checking and pick-up. And, keep in mind, I just broke my rule about posting on a subject that I have no direct experience other than walking by and noticing the lines.
There is also a self-service style storage business located near Termini and a couple of other locations around Rome. https://www.stowyourbags.com/en/shop/rome/
I can confirm the line at Termini is super long. I used www.bagbnb.com and found a hotel across the street from Termini to leave my luggage. It was a good experience.