First trip to Europe: the places that are at the top of every list are there for a reason. They're all special. So I'd stick to the best-known destinations.
Places to avoid: anything that shuts down after the tourists go home. (We have those here too. Think Martha's Vineyard after Columbus Day.)
I agree with staying in the south. Also, unless you're already fairly fluent in a European language, I'd make my first trip to someplace where they use the same alphabet you're used to and you have a decent shot at finding someone who understands a bit of English in most places. Save the tough language problems for your next visits.
Put it all together, I'd vote for Rome. Depending on how long you plan to be there, branch out in Italy: Venice, Florence, all of Tuscany...
And remember: there are few, if any, really bad choices.