Are you wedded to the idea of having just an extra "day or so" in Barcelona before the cruise? Barcelona has tons to see, and a short trip there is very difficult because so many of the top sights require pre-purchased tickets to avoid long lines. It's quite a challenge to know how far apart to book those sight visits, and you can end up with little bits of useless time between them, or not enough time to see them as you'd like. I'd prefer to go to Barcelona at least five days early. The arrival day will be pretty useless, so that would give you four full days to see a bunch of things and get over the jetlag.
There are nearly infinite possibilities between Rome and Nuremberg. It's hard to go wrong unless you decide to do something like visit a string of large cities and change hotels every night. That would be a mistake.
Since you'll be coming off a cruise, perhaps time in the Italian lake district might not be your top priority, but a few days in the Dolomites, where there are lovely easy walks as well as more challenging hikes, could be a great option for a family like yours. If you stay at altitude in a place like Ortisei, you won't have to worry about extreme heat (which is possible in Italy during June). You do not need a car for a short trip to the Dolomites; the bus and lift transportation beyond Bolzano is very good. Bolzano itself is right on the train route between Rome/Bologna/Verona and Munich/Nuremberg. The museum housing the Iceman is in Bolzano, which also has a very pretty historic district that looks quite a lot like Austria. Bolzano can be hot in the summer, and many of its hotels are not air conditioned, so I would stay up in the mountains rather than down in that valley.
Pick up a comprehensive guide book to Italy from your local library and dip into the chapters covering areas between Rome and the Swiss/Austrian borders. See what sounds appealing to you. Think about whether you want to focus on larger, more popular destinations (Florence and Venice) or smaller/less touristy spots. Everyone here who has been to Italy has his or her own favorites.
Alternatively, you could book a flight from Rome to somewhere in Germany and spend all your post-cruise time in that country. This would be most affordable if you are traveling very light, because the budget airlines (whose fares can be extremely low if you buy tickets very early) tend to have confiscatory baggage fees and very low weight limits for cabin baggage. You should pack light anyway, of course, because dealing with heavy bags would not be fun when traveling by train, either.
