Germany/Switzerland/Austria

Just one thought to rental car topic. Be aware that cross-border rentals (drop-off ≠ renting location) can become very expensive. Also have in mind different requirements, e. g. low emission zone stickers in some German cities or Autobahn vignette in Austria.

1) Switzerland is wonderful, but more expensive that Germany or Austria. Still, a great place to stay in Interlakken. Use it as a base for travel within the country.
2) Germany (a)Black Forest- Triberg, Friberg and Titisee are of wonderful memories. Also, you would be near Strassbourg, France, which is worth a couple of days. You might start in this area, then move to Switzerland, then Austria and end in Bavaria (Munich) or do the Romantic Road in reverse and end up in Wurzburg and fly out of Frankfurt.
3) Austria, Innsbruck, Vienna and Salzburg are worth a visit.
4) After Salzburg, just across the border is Berchtesgaden. The Eagle's Nest is awesome, but check for when it closed due to weather.
5) After Berchtesgaden, take in Munich, down to Garmish/Fussen for the castles, then take the Romantic Road up through Augsburg, ending in Wurzburg. (don't miss stopping in Rothenburg ob der Tauber, Dinkelsbühl, Nördlingen and Donauwurth.
https://www.romanticroadgermany.com

Thanks for the replies!

@MarkK: I think our plan for rental cars would be to take trains between a few key bases, then rent a car to explore around those areas.

@geovagriffith: thanks for the great ideas. Do you think it makes sense to fly in an out of Frankfurt?

It might very well be best to fly into Frankfurt, since the Black Forest is not a long drive from Frankfurt. You could stop briefly at Heidelberg on your way down.

If you do the Romantic Road back to Wurzburg, you would be on your way back to Frankfurt.
The other choice would be to fly into Zurich, but then you would have to backtrack to go to the Black Forest.

If you'll be in Munich in early October, then you have an opportunity to go to Oktoberfest. If this is even mildly of interest to you, then I would highly recommend it. Considering the number of people and volumes of beer, the crowd was friendly and quite civil. My wife and I were there in 2018 as part of the RS Best of Europe in 14 days tour and it was a blast. We're planning a return in the fall of 2020.

Here’s how my family explored the Alps and the Black Forest. We started in Geneva to visit CERN. We first rented a car at the airport (Swiss side) and set off to Chamonix the next day to see MontBlanc up and close. We then drove to Bernese Oberland with stops in Lasagne and Bern. We made Wilderswil our base for exploring Gimmelwald, Junefrau and other sites in the area. A few days later, we went to Black Forest stopping in Zurich for late breakfast. We stopped and explored the small rural ‘Badens’ that pepper the area — the taste of fried potatoes and brats served for lunch was unforgettable. We got lost in the hills when our car GPS couldn’t find the satellites to guide us further. That prompted jokes about Grimm brothers stories and why people should always listen to their parents. With the help of locals and our introductory level of German, we made it out of woods fine, bought the genuine cuckoo clock, savored the famous gateau cake, and were back in Switzerland without further delays thanks to no-speed-limit autobahn. In a day, we ventured out to Zermatt and finally back to Geneva.

It might be worth your while to take a look at Lonely Planet Germany, Austria & Switzerland's Best Trips. They have suggested scenic routes. Your library might have a copy