Esslingen's old town is compact and almost entirely walkable. The main sights — the Marktplatz, the two churches, the Stadtmuseum, and the Neckar waterfront — are within ten minutes on foot of each other. The climb to Burg Esslingen through the vineyards is the steepest walk in town but takes no more than 20 minutes at an unhurried pace.
Esslingen has an extensive tram and trolleybus network, one of the most comprehensive in Baden-Württemberg. The trolleybus system is a local curiosity in its own right — Esslingen has one of the few surviving urban trolleybus systems in Germany. However, for exploring the historic centre, you won't need it.
For the Württembergische Landesbühne, which is near the main train station, it's a 10-minute walk from the Marktplatz. Parking in the old town is limited and the streets are narrow; arriving by S-Bahn from Stuttgart and exploring on foot is the most enjoyable approach.