I love both Barcelona and Rome. Almost 40 years ago, I backpacked all over Europe...and every time I make it back to Rome...it is always magical and surreal.
If an easier embarkation port is a high priority..., yes, Barcelona airport is only about 30 minutes. Rome to Civitavecchia is at least 1 hour.
IMHO, starting in Rome is better and is what I would recommend. Spend a week, a few days in Rome...visit St. Peter's Basilica, walk towards the Tiber River, cross the river by Castel D'Angelo, walk to Piazza Navona, then to the Pantheon, then to the Trevi Fountain...the Colosseum, ancient forum, the Spanish Steps. So much more.
With Barcelona being a port stop, you'll be close to the city and see many of the sights...Sagrada Familia, Parc Guell, walk up and down Las Ramblas...and of course everything Gaudi.