travel by train and day trips from around Baden Baden and Mainz,

We would maybe do easy day trips from Baden Baden.

Probably not so easy. I would book in a different town or location. From that spot you must catch a city bus for a 20-min. ride to the station just to reach the station whenever you want to do a day trip by train. That's 40 minutes round trip out of your day. Baden Baden is a poor location for train travelers.

Agree about Freiburg. One excellent hike in the area was from Hintzergarten. But there was a wealth of other possibilities.

My vote would be for basing yourself in Gengenbach in the Black Forest. I stayed there in the Black Forest for 4 nights in 2022 and absolutely loved it. The town is very charming with beautiful half-timbered houses (it's frequently called a "chocolate box" village) and has a beautiful Altstadt, with quite a few restaurants and cafes to choose from. Gengenbach makes a good base as it is right on the main train line, and gives you the options of some interesting day trips, including taking a day trip to Strasbourg, France, which is a quick 40 minute train ride away.

One nice benefit if you stay there is that you receive a KONUS card, which entitles you to free public transport throughout the Black Forest region. There are many charming villages in the region that you can visit along the Black Forest Railway, and there is also the Vogtsbauernhof (open-air museum) that is a joy to visit and easily accessible by train. You can travel to the lovely towns of Schiltach and Halsach by rail, and see incredible scenery on your journey.

The town church (Saint Marion) is quite beautiful, and the medieval gates to the city are really interesting. The Benedictine Abbey is also worth a visit, and behind it are some cloistered gardens that were a joy to wander through. This was one of my favorite places. It was peaceful and inviting and I could have stayed for hours. There were little signs of inspiration scattered around, and some children from a nearby kindergarten playing on one side.

Also don't miss a walk through Engelgasse and Höllengasse streets (near the TIC). Here you will find cobblestones and half-timbered houses, flower pots and cats resting on doormats and quaint dolls in the window. I felt like I had walked back in time when I strolled through these quaint and narrow streets. There is also a nice hike you can take that will get you above the town with some absolutely beautiful views of below. The hike ends up at a church called St. Jakob auf dem Bergle that sits above town. The TIC in town will have maps showing you how to get there. it was a really nice hike and as I said, the views are just lovely!

Gengenbach is right on the scenic Black Forest Railway which mardee mentions, just a few minutes away from the main north/south railway which connects towns to the north like Mainz with Freiburg and Basel in the south. So getting to Gengenbach is easy. You can see on the map below some of the BF Railway stops and the town of Offenburg, where the two railways meet:

https://upload.wikimedia.org/wikipedia/commons/4/4d/Verlaufskarte_Schwarzwaldbahn_%28Baden%29.png

If you stay in the town of Gengenbach as suggested, you will be only a few steps from the railway station for your outings - just as the town map demonstrates:

https://pg.world/upl/ckeditor4_files/1d033e8dc47cc3aca8f58903fa2e5d29.jpg

The terrific Vogtsbauernhof Museum, also mentioned by mardee, has its own railway station on the BF railway as well. (It's very close to the Hausach station shown on the map.) "Gutach Freilichtmuseum" is the stop's name, and it's right in front of the facility, as you see in this photo.

The KONUS card will get you from Gengenbach to Donaueschingen and the other BF Railway towns in between. Villingen is an especially nice old walled town worth visiting:

https://www.villingen-schwenningen.de/en/tourism-experience/experience-vs/places-of-interest/city-district-villingen/

Schiltach was also mentioned. Very nice place, just off the Black Forest Railway. Trains from Gengenbach to Schiltach don't take long. The train leaves the BF Railway in Hausach and ventures up the Kinzig River Valley to Schiltach on its way to Freudenstadt. It's not on the BF map above - so here's a rail map showing that route:

https://de.academic.ru/pictures/dewiki/86/Verlaufskarte_Kinzigtalbahn.png

Were you intereste in visiting Freiburg? It's a pleasant city. Not especially "Black Foresty" but worthwhile. You can take a day trip there from a Gengenbach base quite easily. And of course the trip is free of charge since Gengenbach offers the KONUS CARD.

KONUS Card info:
https://www.youtube.com/watch?v=KkgzMXQ7blQ

So you can count me among the cheerleaders for Gengenbach as both a lovely town and a good travel base.

Thank you Russ, Another Steve, Greg 1262002 and Mardee for the great suggestions. I think we will change our hotel reservation from Baden Baden to Gengenbach. The Konus card sounds like a really good deal. Any suggestions for accommodations? We were hoping for a 3star centrally located near restaurants and bars for a night cap. We will be arriving September 11 from Cochem for 3 nights then to Mainz.
Thank you again, any other suggestions would be appreciated

3-star hotels...

I am not one who pays attention to the classifications when I book as it's about about individual criteria that are particular to me and DW. Also, individuals tend to have different understandings of the rating systems. In case you aren't up to speed on the way Germany's hotels stars are assigned, you might want to have a look at the link below. Click on the gold/blue stars near the top to get at the criteria:

https://www.hotelstars.eu/for-guests/criteria-at-a-glance

Interestingly, elevators are a minimal criterion only at 4- and 5-star properties and only if there are more than three floors, including the ground floor. In practice, elevators are fairly common at lesser hotels - guests who want them just have to look through the property's list of amenities.

Extra-warm weather of late often catches hotels by surprise; many lack a/c altogether, and the star system isn't helpful with this amenity. September isn't usually too bad, but if you are sensitive to heat or just want a/c, you have to hunt a place down that has it, as a rule.