We want to visit several of the towns along the Mosel in September this fall and a debating keeping the rental car a couple more days or using the train. How much problem will ther be finding a place to park the car? Zell, Beilstein....
Beilstein... the town that convinced me that Rick Steves' picked his German recommendations by dart board method.
There's usually plenty of public parking, and it's usually located conveniently by the river.
The main line for trains runs from Koblenz to Cochem to Bullay, and there are regional trains hourly. Beilstein and Zell are short bus rides from Cochem and Bullay respectively. For €20, you can buy a VRM (local transit authority) on/off day pass for regional trains and buses. I went to those places in 2009, and, except for wanting to leave Beilstein an hour before the bus came, never missed having a car.
We did not have any parking problems along the Mosel. Given the choice I always vote for a car unless large cities are involved where a car becomes a burden. The car gives you flexibility especially when the great uncontrollable factor known as the weather rears its ugly head. The Mosel is so calm and relaxed compared to the Rhine and little spots keep popping up where you might like to stop and spend a while. Bernkastel and the Berg Eltz were two high points.
I have to agree with James about Bernkastel. That was probably my favorite town on the Mosel. The bus goes to Kues, across the river from Bernkastel - walk across the bridge. It connects with the rail line in Wittlich, which is just outside of the VRM, so you'll need a Rheinland-Pfalz-Ticket (€24 for 2P). That will cover not only the rail, but the bus. I had lunch in Bernkastel - one of the best meals I think I have had in Germany - a Schnitzel on toast, with white asparagus and mushroom, "mit Käse überbacken". I also sampled three wines, one a very good Riesling, the other a Beerenauslese. Beerenauslese is made from grapes left on the vine until they are overripe and full of sugar. I don't normally like sweet wines, but this was exquisite (@€1,60/oz!) Zell was not as interesting, but I had to see the carved stone statue of the famous Schwarzkatz in the square. Beilstein is actually a very cute little town, with the emphasis on "little". I think there are only two main streets. It doesn't take long to exhaust all the sightseeing possibilities. There is the ruined castle, but it was closed for the season when I was there. And there is a very large church, where I spent the last hours preying for the bus to come.
Use a car. Public parking is well signed, plenty of fellow Europeans visit that town on cars, but the valley is sprawled out with many villages, so don't worry. This is exactly the type of trip a car is most helpful: it allows you to maneuver in relate of weather, to make kick detour or totally unscheduled stops and so on. I'd never visit that area with train, it is just inconvenient.
Tom (OP-Iowa), I trust you understand that Public and Free are not always the same thing! But I thought charges for very convenient public parking in Bernkastel were entirely fair. However, we liked Traben-Trarbach better than Bernkastel-Kues, which seemed very touristy. (I'm also sure we missed many lovely towns.)