Petrified Trees and a lot more - DAS tietz
An interesting thing to do in Chemnitz is visiting the "Kulturkaufhaus DAS tietz". In there, you'll find the Museum of Natural History that presents an artificial volcano eruption every 90 minutes. Furthermore, there is the municipal library and a nice cafe on the second floor. But the best is definitely the Petrified Forest. There are a dozen petrified trees, that look almost like normal trees, were they not 290 million years old... Their surface feels like stone but looks like wood. It's very interesting to learn about their origin. They were found in the northern part of Chemnitz a few decades ago, and it is expected that many more petrified trees are hidden beneath the surface. Thus, Chemnitz wants to apply for UNESCO world heritage status.
If you're in Chemnitz for a longer time, you might want to get the event plan of the "tietz" - there are usually several interesting events like photo exhibitions or literary readings of well-known and less known authors.

Me on top of the town hall tower
Starlight (picture taken from their website)
the new City hall
The Petrified Forest in das Tietz