Classy date: 

Exit at the Rathaus U-Bahn and head to StadtBesen, which is actually at the city hall (Rathaus) on the left side.  Enjoy glasses of regional wine or some light snacks and enjoy the atmosphere.  

*insider tip: If you can start your date before 17:00 on a weekday, pop into the Rathaus proper to ride the amazing "Paternoster" - a continously moving elevator!  Great views of Stuttgart from the upper floors as well.

Then head to the Kunstmuseum where you already have reservations for dinner at The Cube restaurant.  Enjoy fine dining with a beautiful view of the Schlossplatz.  If you're having this date on a weekend and you're a fan of modern art, the museum itself is fantastic as well.

After dinner, walk on the Koeningstrasse towards the main train station (hauptbahnhof), take the elevator to the tower to enjoy drinks and views at Bonatz. And at the end of the night, you're at the train station ready to catch an S-Bahn back home.

Traditional date:

Exit at the Rathaus U-Bahn and head towards City-Besen - a real besenwirtschaft in the heart of Downtown Stuttgart (open September - December).  Enjoy a glass of local wine deep underground in the cellar.

Then walk through the picturesque "Bohnenviertel" (bean quarter) to Weinstube Schellenturm, a 16th century tower fortification, now a beautiful traditional Swabian restaurant.   

Finish the evening watching a great jazz band at the region's premier jazz club, Bix, just a short walk away.