Where to go?

IMO you have more than enough to keep you busy in these 4 places with several day trip possibilities. Traveling to the other countries will eat up precious time.

Since you don't like to stay too long in one place, with 3 weeks I might be tempted to add one more location too. I would probably vote for Barcelona or maybe somewhere else in Spain. Prague, Germany, Denmark all seem to pretty far afield from Paris/Italy and would involve a lot of travel time (although with budget flights that might not be so much an issue anymore).

To minimize the time you waste in transit, how about another city in either France or Italy?

In France, the Alsace region, for example Colmar or Strasburg, would give you a very different experience from Paris – plus German influence (eg, Colmar). Or in Italy, Venice is completely different from Rome and Florence, and is truly unique.

I agree..stick to southern Europe not too far from where you are. I would suggest Spain or Portugal since you are already in the neighborhood!

I really think you need to ask yourself whether you are up for any intra European flights on this trip.
If so, one or two flights ??
Paris >>> fly to Prague/Portugal/Denmark >>> fly to Italy (Rome, Florence & Amalfi) - would involve 2 flights.
Paris >>> train to Barcelona or Munich >>> fly to Italy (Rome, Florence & Amalfi) - would involve 1 flight.
Paris >>> train along the Rhine and thru the Swiss Alps >>> train to Italy (Rome, Florence & Amalfi) - no flights !!

Thanks so much for the responses.

Here's the back story. We've visited Europe back in 2012 we stayed in London and travel throughout England and took the Eurostar to Paris for a day. We were only in Europe for 11 days and we had the best visit and I loved every minute of it. I however would love to see Paris by night which I was not able to do in 2012.

We are fortunate in that we have airfare and hotel stay pretty much taken care of b/c of travel points. So the only travel we'd be spending out of pocket will be the train which isn't too bad and if we do fly (and we've researched) from Naples to Prague or Naples to Barcelona we could fly direct for less than $60 pp.

As it is (on paper) we still have a few weeks before our travel timeline is an option on the airline website.
Fly direct from U.S. to Paris
Fly from Paris to Rome
Train from Rome to Florence
Train from Florence to Naples
Bus or private car to Amalfi
Fly from Naples to ???
Spend 1 day in that destination and fly home the next day.

I've watched Rick Steves 1 day in Prague video which covers so much and I'm certain I can enjoy a day drip in any of these places....that's just how I'm made. I don't need several days in one place, to take in everything there is to offer in the area, while that's ideal it's just not possible. I'm starting to lean toward Spain seeing how it is closer than the others but I really was hoping for something a bit different which is why the other places on my list are exciting to me.

My husband I both do love history and historic buildings and do love beautiful scenery but we also are very good at and very much enjoy checking in to a hotel stepping out onto the streets of a city taking a sightseeing tour if one is offered and then hopping off and spend the rest of the day walking/driving around checking out what we can and stopping to get something to eat at a local restaurant. Then heading on to the next destination with our hearts and minds filled with everything we've taken in which will give us a lifetime of memories. I hope this gives you a better idea of what i'm looking for.

I misspoke above. We are fly direct to Florence from Paris rather than direct to Rome.

Fly direct from U.S. to Paris
Fly from Paris to Florence
Train from Florence to Rome
Train from Rome to Naples
Bus or private car to Amalfi
Fly from Naples to ???

You've got some options to your last planned itinerary that may make travel a little easier.
Fly direct from U.S. to Paris
You could fly from Paris-Orly to Florence on Vueling Airlines.

An option would be to fly from Paris-Orly to Pisa on EasyJet. From Pisa, you could take a short train ride to La Spezia and a local train to Cinque Terre. The C/T is a substitute for the Amalfi Coast and it's incredibly popular and much easier to get to. It's not that far to Florence from the C/T via train thru Pisa.
Then you could take a fast train from Florence to Rome.

It's about 150 miles each way from Rome to Naples and the Amalfi Coast is below that. If you chose to fly out of Naples, you'd end up flying very early a.m. to another gateway airport and wait around until mid day to fly home. Italy doesn't have too many international airports, and you'd do so much better to fly home directly from Rome FCO.

Wow, great info, thank you and thanks for the suggestions I like this idea of flying from Paris to or maybe direct to Rome and making Rome home base. I like the idea of the C/T but Amalfi is one of the main reasons for our trip to Italy that has been on my dream list for years now. Do you think it's doable to take day trips from Rome to visit places in Tuscany and one day trip down to the Amalfi? Or flying direct to Naples and work our way up to Rome so that we could fly out of Rome to our last stop then on to the U.S.?

Fly direct from U.S. to Paris - 5 nights
Fly from Paris to Florence - 4 nights
Train from Florence to Rome - 4 nights
Train from Rome to Naples for Amalfi - 4 nights
Fly from Naples to Barcelona - 4 nights
Return to U.S.

Easyjet flies from Naples direct to Barcelona.
If you chose Prague, or Lisbon, or Berlin you would have flight complications - connections, costs, departure times.