Wow! Great trip!
Changing places/lodgings takes half a day so at least 2-3 nights in each location is (to me) essential to avoid the crazies. I schedule AM, PM and Evening blocks to keep my touring/traveling separate in my own mind.
Lake Lugano and Lugano, Switzerland is right next to Lake Como.
It takes 2 hours to get to Lucerne from Lugano by high speed direct train through the Gotthard tunnel. Or about the same amount of time for a regular train to Altdorf at the south end of Lake Lucerne and then take the steamer on the Lake up to the City at the north end.
Or, take the more expensive Gotthard-Lucerne Panorama Express over the pass and on the Lake.
https://www.sbb.ch/en/leisure-holidays/travel-in-switzerland/panoramareisen/gotthard-panorama-express.html
https://www.luzern.com/en/
https://www.sbb.ch/de/kaufen/pages/fahrplan/fahrplan.xhtml
Maybe Como/Lugano to settle into Europe (2 nights), trip up to Lucerne for 2-3 nights (0ne train/lake steamer day, One art and Museums day in town, or take the lift up to the Pilatus Mtn.) and back to Lugano/Como on the direct train, plus Como to Verona to pick up the N-S train line on day 4/5
https://www.trenitalia.com/en.html
https://www.lefrecce.it/B2CWeb/search.do?parameter=searchOutputViewer&cFID=WGAxSzTQjIaZ
Then north to Bolzano and the Dolomites, or south to Venice. Which one first?
Val Gardena area of the Dolomites is wonderful but there are lots of other places there... It is where I stay when I have been there to ski several times. LaPosta Restaurant in Santa Cristina is wonderful.
https://www.valgardena.it/en/
https://www.valgardena.it/en/bus-schedule/
https://www.suedtirol.info/en
If you like hiking check out Gillian Price's "Short Hikes in the Dolomites." We used it on our visit last summer. It has 50 to select from across the Dolomites.
I understand the attractions of Venice, but in August, and if cost is any issue, I would suggest staying in Padua (45 minutes away by train). It is a university town that is charming in its own right. The Scrovigni Chapel there by Giotto is awesome. And Padua is much more relaxing that Venice. BUT, of course, then you miss the less crowded early and late Venice ambience when the day trippers (like me)/cruise shippers have gone. Your choice, or do some of both.
http://www.cappelladegliscrovegni.it/index.php/en/
The only real "out there"place mentioned is Salzburg, which would add a lot of travel time. 4 to 5 hours each way by train.
https://www.lefrecce.it/B2CWeb/search.do?parameter=searchOutputViewer&cFID=rIonPP595Iyy
BUT you can do Innsbruck, Austria (and Hall, Austria right next door) just north of the Brenner Pass in just over 2 hours by train from Bolzano. And there are gorgeous valleys for hiking/exploration just south of town. Stubaital, Zillertal.
You might want a car for four travelers, but it will need to be big enough to carry all of you and all of your luggage!?!
Everything I have suggested however can be done by train or bus. At least part of the trip's adventure could be trying that out, such as Como/Lugano to Lucerne and back.
Maybe:
2 nights around Lake Como/Lugano.
2 - 3 nights in Lucerne.
2 nights in Bolzano
3 -4 nights in the Dolomites
2-3 nights in Innsbruck
3-4 nights Verona-Padua-Venice
Back to Milan