Take the train or rent a car?

You are going to Zermatt a town that does not allow cars. https://www.zermatt.ch/en/arrival

The public bus is free in Bern and Geneva when you check into a hotel
If you were planning to got to the top of the mountains the Swiss Travel Pass covers mountain excursions Rigi, Schilthorn and Stanserhorn.

Not exactly a wash because the Swiss pass also gives you free
bus rides, museums, boat rides, some free cable cars & discounts
on cable cars/furniculars in Zermatt & Lauterbrunnen area so that's
a major difference in dollars for you to save. Luv the Swiss Pass!

I don't understand why tourists even contemplate rental vehicles in Switzerland. Public transport takes you everywhere you could possibly want or need to go and takes you to places that rental vehicles cannot get to.

Swiss transportation is a delight. It is on time, connections are seamless, the trains and buses are spotless, the scenery is amazing. Just say no to the car. We North Americans ar so conditioned to believe we have to be in control and have a rental car. We love giving up our dependency on wheels and traveling for up to two months at a time by train and the occasional bus. Helps work off some calories, too!

Swiss Pass covers Chamonix? Pretty sure that's in France.

Swiss Pass covers Chamonix? Pretty sure that's in France.

It is, and it does. The Swiss pass covers Martigny to Chamonix, but the Half Fare Card does not. The Swiss Pass covers a few other cross-border lines as well, such as Brig - Domodossola - Locarno.
See the official Swiss Pass map here: https://www.sbb.ch/content/dam/sbb/de/infotexte/uebersichtskarte-sts.pdf

@rjrietkerk, you can use this map for planning- All the red lines are rail lines, and every line has at least one train per hour.

Are you flying into and out of Geneva? If so, Chamonix to Geneva airport is quickest by direct (French) bus. Multiple companies offering this route, just google.

And a car is useless in Geneva and Bern and of limited use in Lauterbrunnen. Lauterbrunnen is the end of the road, and you will be wanting to go up the mountains, that is past Lauterbrunnen, the only road is back down the valley out of the mountains.
You can't even drive to Zermatt.

P.S. Swiss Pass prices and purchase is here: https://www.sbb.ch/en/leisure-holidays/travel-in-switzerland/international-guests/swiss-travel-pass.html

Swiss Travel Pass 3 days CHF 225
Swiss Travel Pass 4 days CHF 270
Swiss Travel Pass 8 days CHF 398
Swiss Travel Pass 15 days CHF 485
Swiss Travel Pass Flex 3 days in 1 month CHF 259
Swiss Travel Pass Flex 4 days in 1 month CHF 310
Swiss Travel Pass Flex 8 days in 1 month CHF 445
Swiss Travel Pass Flex 15 days in 1 month CHF 532

Sweet! I'm totally crashing Chamonix next time.

Train. You haven't factored in parking fees, stress of driving in unfamiliar territory, and stress of finding parking places.

Unquestionably train. With the itinerary you lay out, it's actually faster and far easier to reach those destinations via train and bus. Remember too, that gas and pay parking in Switzerland is insanely expensive. I used to live over there and visit quite frequently and won't get a car unless I happen to fly in and out of Zurich on my way to somewhere else, ie Austria, France or southern Germany.

Definitely take the train or other modes in the excellent Swiss transportation network. As someone else mentioned, you won't be able to take a car into Zermatt. Also, if you'll be exploring the mountain sights or staying in one of the small towns around Lauterbrunnen (ie: Mürren), a car is about as useful as a boat anchor, as it will stay parked during your stay and accumulating rental and parking charges.

Chamonix is easily accessible from Geneva using one of the many Shuttle services. Pre-booking is a really good idea! There is rail service into Chamonix but that will take longer than the Shuttle (although it's a very scenic trip!).