Hi Ken -
In addition to Lucas' comments, I might add that in small towns, very often there is free or cheap parking, although sometimes limited, at the train stations. In "old towns" in any city, it is very hard to park. Train stations are usually near the "old towns."
I repeat Lucas' comments that many tourist destinations are car-free.
I just returned from two weeks in Switzerland, including time in Luzern. I paid nothing to park at my inns in the Emmental, Sugiez, Sumiswald and Hurden.
Due to schedule constraints, I did stay in one night in Luzern, before I went to Weggis where I parked for about 15-18 hours for 3 or 4 CHF in the lot by the valley station of the cableway to Rigi-Kaltbad.
In Luzern, I paid 57 CHF for right around 24 hours, at the Hauptbahnhof. There is no parking close to my preferred hotels.
Unless you have a lot of people in your party, or you have found free parking in Luzern, you have picked the expensive way to travel. If your day trips are to cities or well know tourist destinations, you have picked the inconvenient way to travel.
For day trips, with no luggage, the trains ( and buses and boats) are by far the most convenient, are quite flexible, allow everyone to watch the scenery, and can be less expensive than a car unless you have to buy a lot of tickets for a large party.
To visit the old towns of larger cities can be awkward with a car.. Limited parking, hard to navigate ( one way streets, narrow, many for pedestrians only, sometimes blocked by delivery trucks in the AM when I am in a hurry to get on the road......
My usual pattern is to stay in cities without a car, at the beginning and end of my trips, see friends or do day trips by train for a day or three until jetlag has diminished, then rent a car to go to the countryside and stay at inns for the middle portion.. There, I use a car for hiking. It is more convenient than using public transport mainly because of timing flexibility. So, I use my car to access trail heads for hiking, and some scenic overlooks, and, although bus service or train service can get to those places, the time constraints become more noticeable.
Often, the car sits for days in free parking, while I take a train for a day trip to a city or a place that has good bus or train service. (That may include trail heads, too.)