train Amsterdam to Offenburg and rent a car there

Why Offenburg?
No way is Amsterdam to Offenburg an "nonstop trip". You will have to change at least once (usually Frankfurt). 7 stops between Amsterdam and Frankfurt, 2 between Frankfurt and Offenburg. But, what have you got against stops? Changing trains is also no issue.
I would recommend Karlsruhe or Freiburg-im-Breisgau as better places to pick up your car.
Both places are worth a visit. My suggestion is train to Freiburg, visit the city and overnight, then pick up the car the following day.
Freiburg has good train connections to Paris.

One consideration for exploring the Black Forest, Freiburg, etc. is the fact that the passenger railways in this region are not only extensive and very scenic but also completely free of charge when you stay in one of the smaller villages.

KONUS card

Actually, there are a few (at least one) ICE connections directly from Amsterdam to Offenburg, but going to Offenburg and then driving back to the Rhein involves too much back tracking. Take the train from Amsterdam to Boppard, St. Goar, Bacharach, or wherever, visit the Rhein gorge area (a car is not needed), then go on to Offenburg (a Rheinland-Pfalz-Ticket will cover your travel from the Rhein to Karlsruhe for 28€ for two people). Neither is a car needed for the Black Forest. I know the area; I've spent over three weeks in the area using just public transportation; never needed a car. Buses will take you just about anywhere you want to go - and for far less money.

Yes, Freiburg might be a good initial destination. Or maybe Bacharach. Good to know about the Konus card. We might be able to do this trip to the area and just use public transportation. And not rent a car at all.

I checked the maps again and I see that the regions we want to visit are widespread. Not sure if we'll be able to visit all these without access to a car. These areas include:
the Black Forest area in Germany
the Rhine area Mainz to Koblenz
areas upriver on the Moselle
Colmar area in France

"I checked the maps again and I see that the regions we want to visit are widespread. Not sure if we'll be able to visit all these without access to a car. These areas include:
the Black Forest area in Germany
the Rhine area Mainz to Koblenz
areas upriver on the Moselle"

Do a little research on the German Rail website. Germany has the largest and densest public transportation system in Europe. I spent 5 nights on the Rhein and 4 nights on the Moselle and went everywhere I wanted to go with public transportation. I've also spent, as I said, a little more than 3 weeks in the Black Forest, again going everywhere I wanted to go with public transportation.

I have always found public transportation to be significantly less expensive than rentign a car. With a little research, you can save serious money.

Now, as for the French side of the Rhein, I know there are fewer train lines, so, maybe, a car would be warranted.

dkruz1: The areas you've targeted are generally pretty easy by train or train+bus. I too have spent weeks on multiple trips in these areas using public transport and have also had a car at times. Naturally, there are places that tracks and buses will not take you. Neither Lee nor I know exactly which specific places you are targeting. Maybe you have some inn or winery in the hinterlands in mind. Maybe you are spending 3 months in these regions and are intent on exploring every possible hamlet. If this is not the case - that is, if you're a casual visitor looking to spend several days in each area, you should have no problems filling your days with interesting places and using public transport to visit these areas.

Riding the trains in these regions is so easy, enjoyable, and inexpensive that I doubt I'll ever again be tempted to get behind the wheel there.

Sometimes, trains actually save time over the car. The scenic river route from Koblenz to Cochem takes 58 minutes by car, but the RE train does this same route in 35 minutes. From Cochem, you would need 30 minutes to drive to Bullay (a nice place to begin a hike) because roads follow the curvy river; the train travels through a mountainside tunnel and gets you there in 8 minutes.

The Mosel between Trier and Traben-Trarbach, roughly, is a beautiful, windy, curvy mess where you could spend many hours by bus or by car. The main train route bypasses this section of the river. Traben-Trarbach is reachable via a separate set of tracks from Bullay (which is on the main route) known as the "Moselweinbahn" so no problem getting there. The popular destination of Bernkastel-Kues is reached by taking the train to Wittlich (on the main route) and a bus from there, so that destination is fairly simple too. Buses will get you to other towns as well but will take a while. Below is a map of this part showing the river and the train routes (curiously, the Moselweinbahn train route is absent on this map.)

MOSEL from Trier to Bullay

I have had no bad car rental experiences, but any thoughts I have about renting a car these days have been tempered by the experiences of others who've dealt with shady agency practices. If you decide you need a car you might want to sift through some threads to see what to avoid:

TA thread 1
TA thread 2
TA thread3