Klaffenbach Moated Castle is the prettiest historic excursion on the edge of Chemnitz, a compact Renaissance castle reflected in its surrounding water. It gives the city a softer, older counterpoint to its industrial and modernist associations.
The castle complex is often used for exhibitions, events, and relaxed walks, so it works even if you are not chasing a formal tour. Go for the architecture, the setting, and a break from the denser centre.