How exciting! You have so many options when it comes to visiting Europe.
If you want to visit several countries and really make every second count, I recommend an escorted tour. You won't have to worry about any of the details. You'll get to see all the headline attractions in each country with your own personal tour guide. An itinerary could look something like this:
Another option to see several cities is cruising. The benefit here is that you only unpack once and you have a home base to come back to every night. Plus since meals are all included, it can be a great value.
Luxury cruise lines like Azamara cruises stay late in port or even overnight so you can really make the most of your time in Europe.
Mass market cruise lines like Celebrity, Princess, Holland America, or Norwegian offer great itineraries around the Mediterranean.
River Cruise Lines are smaller and allow you to visit the smaller towns along the major rivers in Europe to get a great sense of local culture.
If you want to go it alone, many of the major airlines have packages available that combine flights and hotel.
If you plan to stay in one place for the whole time, renting an apartment through a website like airbnb.com may be a great option.
With so many options, it just makes sense to have a travel professional guide you through the process of choosing the best option for you and your family.