Basel to France and Germany

You can get a train to all those places from Basel.
Note Basel has two stations, on opposite sides of the city: Basel SBB / SNCF (Swiss and French station, all destinations on left side of Rhine) and Basel Badischer Bahnhof (German station, all destinations on right side of Rhine)

• Schaffhausen -Trains from Basel Badischer Bahnhof. This is a German train line, via Germany. That is a shorter distance than going via Switzerland (Zürich). The Rhine Falls are some distance from Schaffhausen. You will need a local bus. Here a car could be useful.
• Strasbourg, Colmar - Easy by trains. Regular trains from Basel SNCF (French trains, not Swiss).
• Luzern (local spelling) - 1 hour from Basel SBB. You would be mad to do this by car.
• Also: Freiburg im Breisgau - Nice German city. Trains from Basel Badischer Bahnhof.

Other places you could visit for which a car could be useful:

• Route des Vins d'Alsace. Lots of lovely wine villages (tasting and purchase from Vineyards - hic!). Details: https://www.alsace-wine-route.com/
• A drive in the Black Forest

You may want to do the car trips above on first 2-3 days in Basel, then dump the car for the rest.
You cannot drive in central Basel. It is a public transport city. Most hotels in Basel do not have parking.

Be sure to have a day to wander around Basel (on foot). It is a city of streets and lanes full of buildings where you can think "WTF century am I in".
Slideshow: https://www.youtube.com/watch?time_continue=4&v=VGIIbdLu0ug
Or google "Basel Gassen Photos" and click on "Images".

Just to expand on one point. For the French train to Colmar and Strasbourg, you walk outside the Basel SBB station, turn left, And walk along the street for maybe 100 meters to the tracks for the French trains ( numbered 30 and higher). The path is well-signed.

We found it easier to buy those tickets for the Frenchntrains from trainline.eu than from the SCNF website.

Basel is a lovely city to visit. We especially enjoyed a walk through the old town area, and my husband took a swim in the river. If you would like to stay near the SBB train station, take a look at Hotel Gaia and Hotel Euler. ( I did not check for triple room availability, however.)

Thanks for the advice, Lola, scythian and Chris F!

I did not realize parking would be an issue in Basel. This is good to know. We have to stay at a particular hotel for work while we are there so I will check on it specifically.

Thinking maybe Freiburg im Breisgau and some of the Black Forest might be a better day out for us than Schaffhausen and the Rhine Falls. I’ll have to think on this!

All the information about the Basel train station is very helpful as well!

Thanks so much to you all!

If no one else answered it directly...no big deal at all to take the trains anywhere, regardless of nationality. As I recall, I took a French train out of Basel toward Mulhouse and Strasbourg. You're totally fine without the car. Another quick trip from Basel is Bern (on a Swiss train), which is highly underrated, and very few tourists in sight, if any.

Thanks everyone,

We have switched around our itinerary to do Strasbourg, Colmar and Freiburg im breisgau before we drop the car in Basel. Then we will take the trains to Bern, Luzern, etc. from our home base in Basel for the next six days.

We still aren't sure about Schaffhausen and the Rhine Falls. We are all healthy so I think we could do the walking required. I also saw on a website (https://trip101.com/article/things-to-do-schaffhausen-switzerland) that there are Paleolithic and Neolithic caves nearby which is super exciting to me. Can't find any information about if/how you can see these in person though. Hm?

It sounds like you’re returning the car in the same country in which you’re renting it, which saves money. Here are some more notes on the area:

https://community.ricksteves.com/travel-forum/switzerland/stein-am-rein-and-schaffhausen-switzerland

Are you sleeping in Freiburg? I ask because it’s the only place in the BF where you don’t get a KONUS transit card with your room. You can see the BF with or without a car. The driving is slow because the roads are like Vermont or Utah.

I was sorry to miss Le Corbusier’s famous chapel in Ronchamp. But it’s a special interest stop. You have some long drives planned.

Beware of confusing Freiburg and Freiburg im Breisgau

• Freiburg is a city and canton in Switzerland, west of Bern
• Freiburg im Breisgau is a city in Germany, north of Basel and near the Black Forest.

Both are known as "Fribourg" in French.
From Basel station (both stations) you can get a direct train to Freiburg im Breisgau, to Freiburg you need to change at Bern.
If you buy a ticket to Freiburg from an SBB (Swiss railways) ticket machine, that will mean Swiss Freiburg, if you buy from a DB (German railways) machine that will mean Freiburg im Breisgau. If you buy from a human, when they hear your foreign accent, they will ask which one.

This is a photo of the departure screen at Basel SBB: https://upload.wikimedia.org/wikipedia/commons/thumb/5/51/Basel_Bahnhof_SBB_04.jpg/1280px-Basel_Bahnhof_SBB_04.jpg
Half way down, 17:04 departure platform 10, first stop "Freiburg (Brsg)"