Lucerne, Salzburg, Innsbruck

Where on Lake Como? That would effect advice as to itinerary.

If by car, I would pick Luzern first as it is only a relatively short distance from the airport.

Close to Bellagio

Lake Como has two branches. It is shaped like an upside-down Y, with Bellagio on the end of the peninsula between the two branches. I.e. by land it is only accessible from the south, otherwise you need to take a ferry across the lake.

Innsbruck and Salzburg are a long way east. It would take most of a day to get there or back, and you only have 6 nights. Not long enough for both. You could do 43 nights Luzern and then 2 in Innsbruck, then one en route to Lake Como..

The answer really depends on whether you are travelling by car or train. You could cut south via Chur stopping off en route, to arrive at Lake Como from the north.

After Luzern, I'd bite off the big chunk by driving to Salzburg. It's Autobahn all the way. Be sure to stop and buy an Austrian vignette windshield sticker at the border and attach it. You are also required to have an International Drivers Permit along with your drivers license to drive in Italy. Maybe 2 nights in Salzburg.

There are couple of mountain routes from Innsbruck to Lake Como. One is to Landeck, then up the Inn River valley into Switzerland and spend a night in St Moritz. Next day, over the Maloja Pass into Italy. You can drive either down the north shore of Lake Como to Menaggio and take the car ferry to Bellagio, or the south shore to Varenna and take the ferry there.

The other way is over the Brenner Pass from Innsbruck, then to the southwest to Meran(o), then to the Passo Stelvio, one of the most dramatic mountain passes in the Alps.
https://en.wikipedia.org/wiki/Stelvio_Pass
On the other side is the ski resort town of Bormio where you can spend the night. Next day to Tirano and the top of Lake Como where you have the same options as the other route.

I agree with everyone that you should choose Salzburg OR Innsbruck. Which to chose I would say depends on your interest. Salzburg is the more obvious choice, with the Mozart connection, quick hop across the border for Berchtesgaden/Eagle's Nest/Koeningsee sights, Sound of Music, just being charming as crap.

But Innsbruck has it's positives, too. It's closer, for starters. It's less touristy than Salzburg, and while also beautiful feels more like a real place where people actually live and go to University, which Salzburg considerably less so. I really liked being able to get up into the mountains from the center of town via the very cool funicular. Kind of mind blowing to go from the center of town to a ski resort in like, 20 minutes without a car.

If you do choose Innsbruck, the Nala Individuel Hotel is one of the most memorable hotel experiences I've ever had.

Agree about the Nala Hotel!; 2 nights in each are perfect. In Salzburg, I'd visit the castle and do the Rick Steves walking tour on your own. Innsbruck, the Nala hotel and old town. Lucerne, walk the walls, the river bridges and old town. Perhaps a lake tour.

My suggestion would be three nights each in Lucerne and Salzburg, and skip Innsbruck. Keep in mind that you'll be jet lagged for a few days after arrival, so won't be up to full "touring speed". Take the train from Zürich airport to Lucerne and then rent the car before you leave Lucerne. That will save a few days of rental and parking costs.

As I recall, you'll need an International Driver's Permit for Austria, so be sure each driver listed on the rental form has an IDP. As you're driving in Italy, an IDP is compulsory for driving there, and you'll need to be extremely vigilant to avoid the ZTL (limited traffic) areas. Each pass through one will result in hefty fines, which you may not know about until several months after you return home. There are also speed cameras, including the devious Traffic Tutor which measures not only instantaneous speed but also average between two points. Violate either or both parameters, and expensive tickets will follow!

Thank you all very much. Will let you know what we decide!

Another question....Which is the better route from Innsbruck...Innsbruck to Verona to Bellagio or Innsbruck to St. Moritz to Bellagio. Thinking about which is less windy (my son gets carsick easily). Thank you.

Well, I guess that nixes the Paso Stelvio route. I'd do the route through St Moritz, but there is just one pass to go over, The Maloja Pass. Check it out.
https://www.youtube.com/watch?v=EVyKSfHwjfg
Otherwise take the boring route on the Autostrada, the long way around with a lot of tolls.

Thank you. We may just have to get some Dramamine!

St Moritz is beautiful as is the pass to the Italian border. We stayed in an unique airbnb near Stampa for a few days and did a hike from there to Soglio. It's a lovely drive with good roads and you'll probably be taking the Maloja pass slow as there are many busses and trucks.

Take the boring route through the autostrada. I once drove Innsbruck to St. Moritz to Milan, glad to have driven once, but the route is so slow and windy we never felt like doing it again. Actually the Maloja pass is quite easy; what is boring is the long section after the Austrian/Swiss border.

Thank you all for the input! So much to think about.