are you still with us, or have we scared you away with this massive protest against your plans? :-)
"Rethink" may actually be the word.
Let's start out with your kids. I assume that with most families, when they're happy, you're happy. Right?
Now your intention is to "give them a good taste" of Europe. O.k.. So how can that be done in a way that they will enjoy, and that you will enjoy?
Maybe you can start out with the types of activities your children enjoy in general, and then go from there. Now I don't mean take them to the same type of playground they have at home anyways, and buy them the same hot dogs they eat at home. And I certainly don't mean spend all your vacation at Euro Disney.
In Paris, for instance: I am sure they will enjoy getting up on the Eiffel Tower. And I am equally sure they will be bored to death in the Louvre. Maybe there is a boat trip you can take on the Seine and enjoy the town from there. Maybe they even have those small boats where the children get to pedal. I am sure that is going to make a greater impression on them than all the grandeur of the buildings you will be passing by. Go to a bakery together and have your children buy a baguette and a croissant. Have them learn three words of French and give them the chance to apply them.
Now if the grandeur of one city's buildings is lost on most children, why would you want to bore them with the grandeur of three or four cities' buildings? Plus all those even more boring and long tedious train rides in between? They will remember Europe as the most boring place on earth, and never want to return.
One of the greatest intercultural experiences children can have, IMO, is meeting local children. If you can find or arrange opportunities for your children to experience that, I am sure they will start loving Europe and traveling. However, this is not something that will be possible if you are only rushing through places.
Campgrounds for instance are great for that purpose. In our last vacation we watched two maybe six-year-old boys who obviously didn't understand a word of each other's language cautiously approaching each other, taking a long time, looking, staring, retreating to mommy, returning... and eventually playing soccer together. It was a delightful sight to see two children experiencing such a small-scale "international exchange"!
You want to give your children a taste of Europe, and you picked three places. I think basically that's not a bad idea. But for children, it doesn't matter if these places are samples of different countries. It matters if they enjoy the activities, and if in-between traveling doesn't get too strenuous. So why don't you rethink and pick three places that are within easy traveling distance from each other, taking the types of activities your children enjoy as a starting point?
If kids get over-tired and/or bored, they can spoil the whole trip.
Not to mention the trip being spoiled for them. ;-)