Last year, we had a discussion of some interesting places between Zürich and Geneva.
Look in this thread:
Particularly look in my 4th post, and later posts in the thread with details about some of the places on the list given in that 4th post.
The originator of the post "Gusigu" was traveling by train, but a car would work for many of those places. Certainly, though, for the larger cities, getting in and out is much much easier by train, and parking is expensive and not easy to find if you are in the cities.
The following comments may help you think about the choices between rental car and train.
My pattern for traveling in Switzerland for all but the shortest trips is to fly into Zürich, take a day or 3 to catch up on jet-lag ( I'm from the USA, with a 6 hour time change), rent a car and pick up my luggage at the hotel, then drive out of town, typically to Luzern first to visit friends. Even though i know my way around, I avoid driving while jet-lagged. (And, there are trains at least twice per hour for the 45 to 58 minute train ride to Luzern).
As I approach Luzern, I religiously hold to just below the low speed limits between approx. Rotkreuz and Luzern, or I get ( a few weeks later) a bill for 100 or sometimes lots more CHF in the mail from the rental car company and the police. The automated radar speed cameras work very well. They are common all over Switzerland , often in very odd places, and unless you never,ever plan to go back to Switzerland, you'd better pay. And, the rental car company has your credit card number and your address.
For many years i did not try the city driving in Zürich....took my luggage on public transport from my hotel to the rental car location, and drove directly out of own. During the transition over several years, I go lost a lot on the very short route between hotel and Avis. And, if I go off of familiar routes, I get lost frequently in Zürich. Today, with GPS so common ( at a cost of course) it would be a lot easier. However, the intersections come up very quickly, and the streets are far from anything like a uniform grid ( such as in Manhattan.) And, you have to figure out where to park.
I have learned the exact driving route to the Main Station ( Bahnhof) in Luzern, as well as the streets nearby. Once there, I can park for on the order of 50 CHF per 24 hours.
I use local public transport in the Luzern area, but drive to Weggis about 30 km away, to cheap parking by the cable car valley station for my frequent stays up on the Rigi. From Luzern, I drive to other villages all around the region between Luzern, Bern and Neuchatel for my later stays. I stay at inns that do not charge for parking. To visit the larger cities, I take a train from my inn to those cities, or, sometimes, drive to a small town station with inexpensive parking, such as ( one of many ....at random...Konolfingen) leave the car there for a day, and hike, or having parked at (for example) Hasle-Rüegsau station (limited parking) or on the side streets near Langnau im Emmental station or Sugiez Station , take the train into a city such as Bern, Basel, Geneva, Neuchatel.....
The point of those comments is not to suggest specific locations, but to illustrate a pattern and a mechanism for staying at lower cost villages in inns, using the car for inn to inn travel ( which can also be accomplished by public transport, but the schedules at countryside stations are often only once per hour).
The cost of this duplication is manageable for me, because I carry an inexpensive multi-year Half-fare card ("Halb Abo" or "Halbtax") available to Swiss residents or ( currently,at least) those with a Swiss mailing address.
i do it that way, because i have learned not to drive into the cities (take a train) , and often spend a day hiking somewhere where access to the trail heads is not as frequent as i would like by public transport ( take the rental car.)
For you, with a trip as short as yours, if you intend to focus on cities, the trains are the way to go. There are ways to manage luggage. And, you might stay at an an inn in a smaller town along the train route.
There is frequent interurban service all day long into the night between major cities at once per hour, and for many, particularly on the Zürich, Bern, Geneva axis, twice per hour.
or (with less helpful explanation), the direct link -
If you want to spend time in the countryside, on a short trip, a car will be most effective at managing time.
If you have three people, you'll want to balance the cost of a car rental vs. more expensive rail travel, and do the usual trade offs between time, cost and results. In your case, time may drive the setting of priorities.