Best way to get to Bayonne from Madrid, Spain.

We drove, but that’s not best for many people.

I haven't researched this extensively, but my thought would be fly to Bordeaux from Madrid and take the train, or fly to San Sebastian from Madrid, take a bus to Irun, and take the train from Hendaye to Bayonne.

Either fly or take the train from Madrid to San Sebastián. There is one morning high-speed train leaving Madrid at 7:20-am and arriving in San Sebastián at 12:43. Also a couple high-speed afternoon trains. By commuter train ( called the “Euskotren”) transfer to Hendaye. France from San Sebastián. Then from Hendaye, you can take a 35-minute train to Bayonne. San Sebastian’s small airport is outside of Hondarribia, Spain and is just a couple miles from Hendaye, France.
Direct flights from Madrid to EAS ( Donostia/San Sebastián ) take 70 minutes. Indirect flights to BIQ airport near Bayonne take more than 5 hours.
www.TheTrainline.com for train schedules
www.Skyscanner.com for airfares and schedules

Thank you so much for your detailed response. It was very helpful and informative.
All the best!

We did this leg last May as we were coming from Seville, so we were flying. The San Sebastián airport is actually in Hondarribia. We took a taxi from the airport to the train station in Hendaye, bought tickets from the kiosk with our credit card and then took the commuter train 30 min to Bayonne.

If money isn't too tight, you can consider a taxi straight from Bayonne to San Sebastian/Donostia airport. 100 euros on Uber as I type this, it is a 45-minute ride.

There are only 2-3 daily flights from San Sebastian to Madrid, but they are very cheap.
For example, if you take the 9:20 AM flight (arr. MAD 10:30), I would leave Bayonne at 7:00 AM, and you would get to central Madrid by noon. Hard to beat.

When I did this to take the RS Basque tour, I flew from elsewhere in Europe to Madrid, and then from Madrid to San Sebastian. I was pretty exhausted on arrival so took a taxi to the Bayonne hotel. I don't remember the exact price but it was less than 40 euros a couple of years ago.