A visit to the theater should be part of your Leipzig visit.
We can suggest: Schauspiel Leipzig
Famous and worth a visit: Gewandhaus, to be found at Augustusplatz just oposit the Opera house..
Already inn GDR times Leipzig was well known for its Cabaret. The probably best known is Academixer, furthermore there is Pfeffermühle or Funzel and Sanfwut (Mädlerpassage), there are also some smaller ones.
An overview about program / news can be found at Kreuzer!