I am wondering if travel by train from Barcelona Spain to Genoa Italy is possible and easy.
Possible, certainly. Easy, not so much. Other posters can correct me, but so far as I know, you would need to spend many hours and do several changes of train up the coast and around to Genoa. However, apparently Elipsos now offers night service between Barcelona and Milan (possibly stopping in Genoa??). Unless you are sure that you can sleep on a night train, however, and unless you are prepared to shell out some dough for the experience, you might want to stick with day trains, breaking your trip perhaps at a couple of cities along the way.
It's easy if you depart on the right day of the week and can sleep on night trains. There is a direct EuroNight train that leaves Barcelona Franca at 19:38 and and arrives at Milano Centrale at 09:59. Connect to an InterCity train that leaves Milano Centrale at 11:05 and arrives at Genova Piazza Principe at 12:42. However, the night train only runs on Sundays, Tuesdays, and Thursdays.
As Roe said, traveling during the day requires at least five connections. See the timetables at www.bahn.com/i/view/USA/en/index.shtml. After you get summary timetables for Barcelona-Genova, click on "Show details for all" to see the connecting points.
Joyce, I've been struggling with a similar issue. My husband and I are looking for travel options from the Cinque Terre area in Italy (basically right by Genoa) to Barcelona. I think we've decided on taking the train to Milan, which is an easy train ride from Genoa, and then taking an EasyJet flight from Genoa to Barcelona. It turns out that Milan is a big hub for EasyJet, so the total cost of the flight for both my husband and I is only $120 (so, about $60 per person isn't bad at all for a flight). Of course, EasyJet finds ways to tack on additional charges left and right if you have to check bags or go over luggage weight limits, so look at that carefully.
Although we plan on traveling with a 21-day Eurailpass, it still seems worth it to incur the additional expense of flight tickets from Milan to Barcelona to make a substantial savings in time (the flight is only 1.5 hours) and hassle (because of all the train transfers required on the long trip between Genoa and Barcelona).
Hope this helps!