The S-Bahn train is frequently run from Terminal 1 and will take about 30 minutes to reach the city center. Tickets can be bought from machines above the railway platform as well underneath the airport but these tickets must be validated at one of the orange boxes located on the platform before you are able to board the train.
Alternatively, there is a regular bus service linking the airport to various points all over the city. The buses depart from the airport at Terminal 1 Arrivals.
Taxis from Terminal 1 (Level 3) in Departures will get you into the city in about 20 minutes and are in operation throughout the day and night.
If you do not wish to rely on public transport or taxis while staying in Stuttgart then hiring a car will give you the freedom of travel. Car hire companies can be found at Terminal 3 Level 2.