Paris to Cinque Terre with teens

I'd probably make a stop in Nice for a few days. It's easily reached from Paris using the high speed TGV trains. The trip from there to the Cinque Terre (using Monterosso as an example) is about five hours with a minimum of two changes. Time will vary if your destination is one of the other towns.

No simple answer, I think. It depends on larger issues about your trip, ie where else are you going on your trip, how long do you have, what are your wider plans, what are your interests, etc.

That's assuming that you are not just flying to Europe with the intention of only going to these two places, the trip is not just exclusively to Paris and CT, with no place else included (which some would argue is not a logical grouping).

We have 3 weeks. Paris, Cinque Terre, Rome, Venice, Vienna

Consider flying nonstop from Paris Orly to Pisa on Easy Jet. (RyanAir flies from Beauvais, which is much further from Paris, so not recommended.)

We have 3 weeks. Paris, Cinque Terre, Rome, Venice, Vienna

That's helpful.

In that case, I'd say your trip plan is actually centered on Italy (mostly northern Italy), with side trips to Paris and Vienna. Next I would do a reality-check...do you really have 3 full weeks, on the ground in Europe, not including your travel days? Hopefully yes, because you have a lot on your plate. With 3 full weeks, it's doable, but if you start adding more stops (which will be a great temptation) then it could get stretched thin. Have you been to Italy before? You're going to skip Florence. How about Tuscany? Lake Como and the north (you'll feel like you're almost next door, but adding places adds travel time and days). Of course, many people could easily spend 3 weeks just in Italy.

If you really want to focus on the places you list above, Paris is the obvious outlier. I wouldn't "break up" with a stop between Paris and CT. I'd simply fly between Paris and someplace in Italy, it's more efficient with your time.

Maybe fly in to Paris for a few days there, then fly to Rome. Train to CT, train to Venice. Train to Vienna. Fly home from there.

But...I suspect that "other stops" could be added to your list. Still, I'd do it as described immediately above. You're going to need to resist the temptation to add everything, or your 3 weeks will fill up (and overflow) quickly.

I agree with David that even with 21 nights in Europe you are already heavily scheduled, especially when you add in travel days. Flying from Paris to Pisa or Rome is much more practical than taking the train and adding additional travel days and overnight stops.

I'd probably go south through Provence, French Riviera, Italian Riviera, then Cinque Terra. You can get to the Cinque Terra by train from Genoa then leave through La Spezia on your way to Rome.