As above take the train
https://www.frankfurt-airport.com/en...rman-rail.html
The long-distance train station at Frankfurt Airport
The long-distance train station links the airport with all of Germany and the rest of Europe.
The long-distance train station operated by German Rail (Deutsche Bahn) at Frankfurt Airport makes it simple to get there by train from anywhere in Germany or Europe. Information on schedules and fares is available on the website of German Rail.
https://reiseauskunft.bahn.de//bin/q...RAPORT-EN_LZ03
https://www.stuttgart-airport.com/getting-here/by-train
The Stuttgart Central Railway Station is part of the ICE, EC and IC network and located in the centre of the city. You can continue your journey to the airport from here with the municipal rail lines (S-Bahn) S2 or S3 (no changes necessary).