You have two choices.
Grimaldi Ferries Prestige runs an overnight ferry service from Rome (actually the cruise port Civitavecchia) nightly except Sunday. Fares for single cabins range from 139 euros up to 353 euros depending on the time of year (August is most expensive) and whether the cabin is inside or outside (with a porthole or window). Doubles, triples and quads are available at much less cost per person The real deal are the Pullman reclining seats which sell for between 44 and 79 euros depending on the date of sailing. Easily the cheapest way from Rome to Barcelona. The two ferries are high speed and have many of the facilities of a small cruise ship including restaurants, bars, disco, swimming pool and shops. They are named the Eurostar Roma and the Eurostar Barcelona. The sailings are at 7 pm (19:00) each evening. Arrival in Barcelona is at 14:00 (2 in the afternoon). The website is www.grimaldi-ferries.com
If you are in Florence, you can sail from Genoa. The company here is Grandi Navi Velocio sails every two or three days from Genoa at 21:30 (9:30 pm) and arrives in Barcelona at 11 am. Some sailings stay a day and overnight in Barcelona because they market a three day mini cruise on this ship as well as running it as a ferry. Grandi is actually a subsidiary of Grimaldi (above). Cabin fares are slightly cheaper than from Rome except for the Pullman Seats (which are called "sleeperettes") and which are more expensive ranging from 75 to 95 euros. Maybe they are fancier. The ship is the Mv Fantastic and has similar facilities to the Eurostar sisters described above. www1.gnv.it/English/ should get you the site.
The Mediterranean is usually smooth as a bath-tub during summer but not always and not in spring and fall so be safe and bring a seasick pill.
Have a great trip.
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we went from Barcelona to Pisa with ryanair. It costed 150 euros for 4 of us. We took any easy early morning scenis bus to Girona airport and in one hour we were in Pisa. Quick, simple, no crowds, just have to catch the 6:00 a.m. bus in Barcelona at Estacio del Nord bus station which was a 2 minute walk from our apartment. It if isn't an early morning cab will only set you back a couple of euros. So if you're willing to get up early, and go to Pisa...we stayed in Lucca....which is absolutely wonderful and did Cinque terre, which was drop dead gorgeous and then we took the train to Rome. the train costed 34 euros per person. Hope that helps