3 week quick paced itinerary ideas

The RS Best of Europe 21 day tour could be a blueprint. Then just add "Northern Italy" as desired.

( https://www.ricksteves.com/tours/europe/europe-21-days )

If the goal is just to see a lot of countries, the Benelux is a good start. You can then visit France, Germany and Switzerland on your way to Italy. Denmark and England can also be added without too much trouble.

That's kind of a fun assignment. If the following is too much I would cross off Berlin.

How about Berlin, Prague, Bratislava, Budapest, Vienna, Salzburg (with a day trip to Munich) and down into the Dolomites, Venice, Lake Como

Assuming you’ll have a car and specific places don’t matter at this point, France - Luxembourg -Belgium - Netherlands - Germany - Austria - Liechtenstein - Switzerland - northern Italy. Not a trip I’d want to make now, but did a similar one back in 1980 when we were young, but started and ended in Spain. That trip was about 20 days long.

Especially for first timers/teenagers, I really like Rick's priority advice:

5 days: Paris, the Alps
7 days, add: London
10 days, add: Rome
14 days, add: Rhine, Amsterdam, Haarlem
17 days, add: Venice, Florence
21 days, add: Cinque Terre, Rothenburg, Bavaria
28 days, add: Bath, Salzburg, Hallstatt, French Riviera

So, start in London (w/Bath) -> Paris -> Amsterdam -> Rhine -> - Rothenburg -> Bavaria -> Swizterland/Luaterbrunnen -> Cinque Terre - > Rome -> Florence - > Venice

Of course, you can add and remove, prioritize and customize if your family has a special affinity for any particular region, whether for the language, culture, cuisine, weather, or whatever.

For a more specific itinerary.

Start in Copenhagen, add a day trip to southern Sweden. Then Hamburg, Amsterdam (or some other place in the Netherlands), some place in Belgium, Luxemburg-Strasbourg-somewhere in Switzerland-Italy. Too rushed for my taste, but if you don't mind that it should be doable without to much hassle.

Do you realize that with multiple countries you will have to keep up with multiple ever-changing covid policies and procedures? Are you ready to set up covid testing for all of you each time you cross a border, not to mention tracking forms, perhaps getting country-specific QR codes, etc.???
We actually went to Switzerland this past Sept, and the only countries' rules we had to keep up with were the UK [flight transit] and Switzerland, and that was work. I'm glad we did the trip, had a wonderful time, but you can't really unplug anymore because of the possibility of changing rules.
And- there's the CDC changing various countries' status [recommendations only]. However, this COULD affect your travel insurance, [policies vary] if you travel into a Level 4 country.
I love to travel, and already have a trip scheduled for late April, 2022, but we are not crossing multiple borders.
I'd recommend reading through the past few months posts here on the Forum, to better understand the stress of meeting the ever-changing covid rules, especially when the rules change two days before your trip.
But I do hope you plan a great trip and have wonderful family adventures!

We can hope that the COVID situation is much improved by July. Things do change fast--for better and worse. I'd be more concerned that rental car prices will still be high due to the shortage of cars brought on by the pandemic. Also, it is typically much more expensive to return a car in a different country than where it was rented. Also, I don't have a lot of information on this because I've never rented across country lines, but many countries have vignettes which are kind of like a toll except it is some kind of pass that you buy for the car. I personally would look at itineraries/routes that you can do by train or bus. Perhaps just rent a car for Northern Italy.

A vignette will be needed for Austria and Switzerland. One country’s is reasonably priced while the other is expensive. I forget which is which. Unless you rent the car in one of those countries you purchase the vignettes at the border crossing. The vignettes allow you to drive on the highways in the respective country, so they’re more like a road usage tax rather than a toll.

I had been looking at rentals in Spain and Portugal and they were lower than other countries. I'm currently looking for a rental car for Sicily for April/May, and I'm get rates of $50/day at best. Some cities, I was seeing over $1100 for about 14 days.

In this market, I would disagree that rates start low and get higher. Sometimes. It really depends on location. In the past, that was true.

I would look at the Balkan States (Croatia, Montenegro, Bosnia & Herzegovina. Close together, varied in what they offer and easy COVID rules. Then maybe Wizzair to Budapest and Wizzair on to Italy. Those connections run about $75 each. A bit of all sorts of cultures. If you change in Istanbul, you can add that to the trip

Fly into or out of London and take the Eurostar to Paris (2h 30m). While in London you can take a day trip to Bath (1h 30m).
From Paris take a direct train to Brussels (1h 30m). Make sure you take a day trip to Bruges (1h).
From Brussels take a direct train to Amsterdam (2h).
From Amsterdam take a nonstop flight to Berlin. Easyjet has fares for less than $50. Pack light so you don’t have to check baggage. When you’re spending three weeks abroad do laundry. This is easy if you stay at an Airbnb. The farther out you buy your Eurostar train tickets and book travel on low-cost air carriers the cheaper it is.
From Berlin take a direct train to Munich (4h 15m) and explore Bavaria and take a day trip to Salzburg from Munich (2h).
From Munich take a direct train to Verona (5h 30m). You can fly home from (or fly into) Venice (1h 30m) or Milan (1h 15m) from Verona.

If I were you, I'd ask the wife and the kids to each pick a first and second choice for a city, then I'd see how to accommodate at least one from each.

A vignette will be needed for Austria and Switzerland. One country’s is reasonably priced while the other is expensive.

They are both reasonably priced or expensive depending on your point of view.

Switzerland costs CHF 40 for 14 months (December the year before - the whole next year - January the year following) so even though it is valid for 14 months, at 365 days in a year each day is only CHF 0.11 per day. That would be CHF 1.09 for 10 days.

The 10 day Austrian vignette at 9.40€ is 0.94€ per day or it would extrapolate to 343.10€ for a year at that rate. (A year one is actually 89.20€).

They are the same price if you are a tourist or a resident and need one for your car.

To me it is horses for courses. All in the point of view.

I think Pat is right: If you don't want to spend a lot of time planning alternative destinations and keeping up with shifting rules and infection rates, don't design a country-hopping trip. I understand the desire to travel through multiple countries; I've taken a lot of trips that did just that. But it's an iffy plan right now. I've been driving myself crazy trying to research the Nordic and Baltic countries (some of which were closed to Americans for most of the last 18 months) and working on alternatives like northern Italy. I'm retired. I can't imagine tackling something like that if I had to work for a living.

There are countries that have great regional variety such that you could have many different experiences without crossing borders. I'd give serious consideration to planning a trip within one of those countries along with one or a few cross-border jaunts that could be included if the starts align. Here are three off-the-top-of-my-head ideas:

France: Paris - Normandy - Brittany - Loire - Alsace - French Alps - one of the wine regions - Riviera - Provence - Dordogne. That's way more than 4 weeks' worth. Pick those that most appeal or substitute your own ideas. If concerned about hot weather, maximize time in Normandy, Brittany and the Alps.

Italy: Venice - Dolomites - one of the lakes - Bologna and foodie side-trips - Florence - small-village Tuscany - Rome - Amalfi Coast. Extra-credit substitutions: Puglia or Sicily. (Sicily really needs at least 2 weeks by itself.) Caution: Weather likely to be seriously hot from Bologna south; might be hot everywhere except at altitude in the Dolomites. That adds up to more than four weeks.

Spain: Barcelona - Pyrenees - Basque Country - Galicia - [from this point on heat is a near certainty; proceed at your peril] Madrid - Toledo - Cordoba - Seville - Granada. Again, that's more than four weeks. Hot weather is quite unusual at altitude in the Pyrenees, in the Basque Country and in Galicia. Close to the northern coast all the way from the Basque Country to Galicia you're much more likely to have cool, overcast weather (and some rain).

If you're cringing at all my references to summer heat, you could design a trip in Germany, where you'd be very unlikely to run into unremitting heat for most of your trip--though it would be reasonable to expect some hot days. I'm not familiar enough with Germany to pin down specific stops. Include at least one Hanseatic city in the north, the Rhine, Bavarian lakes and/or the Bodensee (Lake Contance), the Spreewald, Berlin and the eye-candy town of Quedlinburg.

I'd plan on fewer countries but more locations within those countries for two reasons. One of course is Covid and dealing with varying restrictions. The second is logistics for traveling with four people which is easiest by car and economically feasible compared to train or plane assuming you are keeping the car in one country. France and Italy are great for touring by car, except for major cities of course. Having a car also allows you to stay in out of the way places that are far less expensive. I'd include one flight at most and a few three hour trains.