The easiest would be to do the complete trip by train (the fastest connections take 5 1/2 hours, one change).
By plane, the easiest would be to go to Frankfurt and take the train there at the airport (1 1/2 hours to Baden-Baden, one change). Strasbourg airport or Stuttgart airport to Baden-Baden by train both need two changes. There is also the option to take a Flixbus bus from Stuttgart airport to Karlsruhe (according to the timetable faster than the train, but may suffer huge delays because of road traffic), and change to the train there.
Fly to Frankfurt hire a Porsche and drive to and through the Black Forest. This will be a trip you will never forget. Ok it will cost you the best part of €1000 just for the car but i garuntee you it will be the experience of a lifetime.
From: Amsterdam Centraal
From: Entzheim Aeroport
From: Stuttgart Flughafen/Messe
With the latter 2 you may select "only local transport" to see cheaper tickets.With tickets only good for regional trains also discounted tickets are flexible. Using a Baden-Württemberg Ticket from Stuttgart Airport - this ticket would also cover the buses in Baden-Baden. And here you have also the option to travel by train through the Black Forest.
Note that Baden-Baden train station is in the suburb Oos. Baden-Baden town is so 5 km away. Frequent buses. Or taxi.