Helpful tips:
Fly in a day early, or two. It helps with jet lag, and more importantly, gives you a cushion in case your flight is delayed. We've been as much as 24 hours delayed on transatlantic flights before. We've been delayed more often than not when going or returning from Europe.
Pack a full outfit in each traveler's carry on, and consider cross packing some clothes in each other's luggage, in case one suitcase gets lost/delayed. My suitcase went missing in Iceland and I had to wear my mom's clothes for 2 days before it finally caught up with me. My son's suitcase went missing for 8 weeks on our flight home.
Get second seating for dinner. That way there's no rush to get back for dinner. Plan on an afternoon snack/light meal to keep everyone going.
Port days are LONG, and the Med is HOT (though hopefully not as bad beginning of June as later) Not really a tip, but be prepared. You may end up wanting to do laundry. We would get back, shower, then wear the next day's clothes to dinner.
With a group of 6, private tours may be a good option. I look at cruisecritic and tripadvisor and contact several companies, then pick the one that seems the more responsive to our requests. We travel in a family of 8 - 4 kids plus grand parents. Google "Port excursion in XXX" and you'll get ideas of what's out there. My family loves food tours. Rome is awesome, but it is a long drive there (1-2 hours), in a lot of traffic usually. Consider carefully how much you want to do with the kids in tow. Private tours allow you to stop for an ice cream break, adjust the pace, etc.
I loved Lucca as well. The bike ride around the city walls is easy, wide, and flat. I think even a 7 yo could do it. You rent bikes at the train station across from the from front gates, and the bikes came with locks, so you could stop and go into the old town at several different locations.
Have a great trip!