If going through immigration, keep in the line to the left which should then later split providing another immigration agent.

There are also direct buses into Tuebingen, our by far favorite German town. just quintessential Germany, from the airport besides taking the S-bahn into central Stuttgart.

Perhaps the tourist information office at the airport or across the street from the Stuttgart train station still has a map with a suggested short 1-3 hour walking tour of central Stuttgart. Make sure to visit the Stiftskirche and the Old Palace which is actually a Renaissance castle with a lovely courtyard. There is also the horse staircase up which royalty rode horses up to dinner and then later further on up to bed. The rams on the clock in the corner of the courtyard butt heads to mark the hour. The building houses one of Germany's oldest custom-built Protestant churches, hosts the excellent Wuerttemberg Museum, and also a Staufenberg memorial/museum to the perpetrators of a Hitler bomb plot because the family lived in this palace when their father was an advisor to the King of Wuerttemberg. The nearby Markthalle is worth visiting for sights and smells of gourmet food from throughout the world, the music instrument museum next door on Schillerplatz I believe is free, The Rathaus (city hall) has a paternoster (continuous doorless elevator) which you can ride. Enjoy your visit!

If you choose to stay at the airport, there is a small aircraft museum on the upper floor. Perhaps there would be an event of interest at the Messe next door. And for an excellent lunch, go to the Moevenpick Hotel across the road and eat at their Trollinger restaurant which is excellent. Their Trollinger red wine from the Stuttgart city vineyard is excellent, as is their rare white Trollinger, just more than excellent. Now I'm getting thirsty!

Thank you both for the replies. We will be arriving first in Zurich so we will not have to go through immigration in Stuttgart. I am looking forward to spending some time there.