12 days in Southwest Germany

Perhaps some of the later replies to a post that I put out a while back may help you on the Mosel piece and cycling.

Despite the title of the post (below), there is a LOT in the later replies on the Mosel. From what I learned, we are now planning a five day trip from Trier to Koblenz, with an extra day in Koblenz to turn the corner and go to Marksburg Castle and/or Boppard and back to Koblenz.

https://community.ricksteves.com/travel-forum/germany/rhine-and-beyond-itinerary-ideas

4 nights in Traben-Trarbach: I imagine that means about 3 days for sightseeing, with the first day given over to the journey to T-T and perhaps some stops on the way.

You probably have something in mind already for sightseeing on those 3 days since you chose to book four nights there, so it would be good to share whatever priorities you have at this point.

Cochem and Bernkastel are common small-town targets, with the most to see/do and plenty of spots for food/wine. Trier (much larger) is interesting but doesn't sound like what you have in mind.

From T-T > Cochem takes roughly 45 minutes, slightly less time by train than by car.

To Bernkastel, it's a direct trip on the hourly bus from T-T (#360) which takes 45 minutes, about 15 minutes more than if you had a car at your doorstep. But what I'd recommend is the Kolb River Cruise to Bernkastel, which we took a couple years ago. It's quite a lovely way to see the river. Then use the bus to return (or the cruise boat again, if you have the time for it.) T-T > Bernkastel cruise schedule here, look at bottom-left corner:

https://www.moselrundfahrten.de/cams/clients/kolb/media/downloads/file_1.pdf?1713965549

So I don't see much need for the car on the Mosel. I've stayed in different towns there on multiple trips and never felt a need for one. But maybe you have some sights in mind that are unusual?
 
Burg Eltz Castle can be combined with a visit to Cochem. You can take the train to MOSELKERN and hike in from there (1 hour, the traditional way.) Or you can take a train to Hatzenport and a connecting bus to the "Burg Eltz, Wierschem" stop; I could see the morning in Cochem with an afternoon jaunt to Eltz (13:24 - 14:13 by public transport for example.) Maybe have dinner in Hatzenport or in Cochem on the way back to T-T.

Getting to Heidelberg from Lindau and then to T-T is also quite doable by train, and it sure beats all that driving, IMO.

Here is a railway map just for the Mosel River train stops:

https://i1.wp.com/cityscape-travel.com/wp-content/uploads/2020/10/RPBahn2.png?fit=1024%2C1171&ssl=1

I should mention also the hike between T-T and Bernkastel. I did this one about 20 years ago with a pack and using a vague trail map that I tore out of a guidebook. It was a memorable hike, and I remember I was out there for 2.5 hours, but I don't remember my route exactly. This page from a local TI site has a route that might be the one I took or might not. Says 2 hours...

https://www.mosel-karte.de/en/tour/hiking-trail/from-bernkastel-kues-to-traben-trarbach/3158902/#dmdtab=oax-tab2

by car or should we travel by train?

Trains are very often recommended: https://int.bahn.de/en

One way car-rentals are often expensive.

For driving a car in Germany you will need driving permission: for this check for each driver validity of foreign driving licences in Germany plus evtl. needed IDP.

We would like to avoid driving on the Autobahn. Is that possible?

Possible. For the long distances drive Bundesstraßen (federal overland roads). In maps their road numbers always start with "B" on yellow-ground road signs, e. g. B31. Use the setting "Avoid Autobahn" in navigation systems for route planning and in practice do not follow blue Autobahn signs (e. g. signs 430 or 440).

Lindau is very popular, but I find it small and crowded. Off the island the area immediately around it is not as nice. If you can, one day when the weather is clear, go to Dornbirn, south of the lake in Austria, and take the cable car up the mountain. There are good hiking trails there and eating dinner while looking out over the lake while the sun goes down is not to be missed. You would be wise to have reservations for a table. You may also want to look into the possibility of taking a Zeppelin ride over the lake.

If you plan to go to Mainau, from Lindau, go to Meersburg and then take the ferry to Konstanz. From there take the bus or a taxi to Mainau.

Be advised that traffic on 31 (Germany) or 13 (Switzerland), around the lake, travels at about 30 MPH, slower in town, slightly fast out. It can be a long, slow, commute.

Travel from Lindau to Heidelberg is approximately 4 hours by train. Driving without using the autobahn will take twice that.

In and around Heidelberg you have a lot of things to see and do, most easily reachable via train. The wine region here runs north-south, not along the river, but along the escarpment of the Pfalz forest. Look for the towns of Neustadt (on the Winestrasse), Bad Durkheim, and Landau. These are very nice, and worth visiting. For even less known spots consider Deidesheim or Freinsheim, small villages where you're not likely to see, or hear, an American tourist. Weingut Dr. Burklin-Wolf in Wachenheim is a favorite place of mine to sit and have a glass and look out over the vineyards, and in the same town the Metzgerei Hambel is world famous. The Deidesheimer Hof is an excellent place to eat, and the 1911 wine bar on the same plaza is excellent. Bad Durkheim usually has a flea market on Saturdays at the Saline (you can't miss it), and the Casino Beer garden is a great outdoor place for lunch.

Going the other way, from Heidelberg, you may want to take a day and visit the small town of Nekargemund (easy by train), it's picturesque, and rarely visited. The castle at Dilsberg is right there and, at least, as good as the one in Cochem. There's a couple other castle ruins visable from the town.

Train from Heidelberg to Trauben-Trarbach is over 3 hours. Driving without the Autobahn about the same. (You may want a car along the Mosel.)

From T-T you have a number of options, and a bicycle and the train are good for most, as long as you stay along the river. The bike paths are wide, smooth, and separate from the vehicle traffic. They can get crowded. You can easily go from town to town, and then to return just get on the train with the bike (look for the properly marked cars).

I like to have lunch in Bullay, sitting in the small park along the river. While the Mosel is wine country, south of T-T is the Brauhaus Koster Machern, which I recommend if beer is more to your taste. They also have good food and a nice outdoor dining area. It gets very crowded on weekends. While most visitors head to Burg Eltz, I recommend to pass on that, and instead take the train to Trier, where you'll have fantastic Roman ruins, a great cathedral, and a lot more to do. You can spend multiple days there and not see everything. The Landesmuseum has a fantastic collection, the most Roman mosaics outside of Pompeii, and there's a nice cafe in the park next door that's good for a quick meal. The wine caves in T-T are not open every day, so make sure they're open (evenings) while you are in town. If you go to Cochem, try to be there before 10 am, as that's when the tour boats and busses start to arrive. Get some pastry at the Lutz bakery and go sit along the river and enjoy before the crowds arrive.

A bit late to the party, but still -
Where at the Lake are you staying? I am mostly familiar with the west, where Meersburg, Überlingen, Konstanz are nice spots to stroll along the quai with some ice cream. Easy to get to the Hinterland for a nice walk. Traveling between cities by train is fairly easy.
For long distances by car the autobahn makes sense - but in the end it is just a highway system, no need to go 150mph if you do not want to…

I really liked having a car in the Mosel Valley. We only had two nights there, and a big part of our time was spent just driving around and taking in the incredible views. Parking could be an issue in T-T, so make sure your accommodation has a space for you.