Berlin has dozens of venues for performances. There's lots of great theatre, but most of it is German-language; it's getting better, with English plays or surtitles in some theatres.
Berlin is the world capital of contemporary dance; look for shows by Sasha Waltz, perhaps the best choreographer around. The Staatsballett Berlin is the main classic dance company. Tanzraumberlin magazine (www.tanzraumberlin.de), available at the venues, lists all dance events.