May should be OK for visiting Rome, not too hot yet. From there you could go to Florence (Firenze), which is one of those places which need to be visited. If you can't do May go the second half of September, but by all means avoid the summer in Rome. Way too hot and humid, way too many people everywhere.
Paris is nice too, but it's more about grandeur (and damn expensive) than anything else. Moreover most Frenchies hate americans, Parisians hate everyone else anyway, and no matter how good is your French, they'll know, more or less with the same accuracy a London snob would know you're not of royal breed. Well, just kidding, or not.. well, Paris deserves a visit too, but the car drive up Ireland's west coast sounds very romantic, and more relaxed.