First Time in Switzerland, last minute...

Normally the excellent train system is the way to go in Switzerland, but since you have an 8 year-old with you a car may be more practical. I have not driven in Swiss cities so I don't know how much of a hassle it will be to find parking in Bern, Basel, and the like.

Bern has an Einstein museum and a modern art museum focused on Paul Klee. Both are worthwhile if the subject matter is of interest to you.

I don't know how much time you have after Zurich, but a few days spent in the Bernese Oberland mountains is unforgettable. Specifically, consider Lauterbrunnen since you'll have a car. You'll be right in the middle of the alps. A 20-minute walk from Lauterbrunnen is Trummelbach Falls which is a spectacular cave system drilled out by glacial waterfalls inside the mountain.

After Chamonix consider stopping in Montreux for a morning or afternoon to see Chateau Chillon. It's a great castle sitting on the edge of Lake Geneva. They have lots of events aimed at kids, especially you son's age. He'll go nuts!

Egmar, Notre Dame du Haut is not in Switzerland, it is in Ronchamp, Haute-Saône (France). You will need a car to get there.
The Vitra design museum is not in Switzerland either, it is in Weil am Rhein, Baden-Württemberg (Germany). It is reachable by bus from Basel (Weil am Rhein is a suburb of Basel), but is probably easier to get to by car.
And Chamonix is in France, Lake Como and Milan in Italy. This "Swiss trip" doesn't have much of Switzerland in it!

Basel however you cannot drive into the centre of. There are car parks on the edge of the centre (pay), and you then get in to the centre by tram. Basel is tram city, lots for your son to see.
Architecturely, Basel has lots to see, old and new. For the new stuff, see this webpage: https://www.basel.com/en/Sightseeing/Modern-architecture
For the old stuff, see here: https://www.basel.com/en/Sehenswuerdigkeiten/Die-Basler-Altstadt
For castles, see here: https://www.basel.com/en/Sightseeing/Fortresses-castles

Bern is similar, no cars in the centre but trams instead.

Ha!
Yes, perhaps my "Swiss" heading was a bit off...I did know we'd be crisscrossing those borders...but since we're starting in Zurich I suppose I consider this our Swiss trip.
Thanks for all the information, much appreciated.
Sounds like we'll NEED a car for parts, so best to rent a car and drive.
Thanks again

Check rental rates carefully; there's usually a very high fee for dropping the car off in a country different from the one where you picked it up.

You'll need a Swiss vignette to use the highways in that country, which I think will be unavoidable. Folks have reported that cars rented in Switzerland already have the vignette affixed, so you avoid the out-of-pocket cost of 40 Swiss francs.

Driving the car into France or Italy is no problem. But if you pick up the car in Switzerland, then also try to drop it in Switzerland, to avoid fees to have the car returned to Switzerland by the agent. Hertz and Europcar both have offices at Taesch, near Zermatt, or more agents are located in Lugano, near Lake Como (local buses connect Lugano with Menaggio).