You can take the train to Krakow. It's about 6 1/2 hours with a change in Katowice. Krakow is awesome! Visit the salt mines. You could visit Auschwitz if you are in the mood. Be forewarned. It's obviously very depressing.
If you prefer to stay in the Czech Republic, the following places are really nice to visit:
Cesky Krumlov - lovely small city with an amazing castle and a river
Karlovy Vary - this is a spa town in the mountains with underground springs; very different architecture from Prague
Telc - another very pretty small town; town center has colorful buildings; a wonderful castle; and ponds on either side of the town.
I have heard Olomouc is nice but I haven't been there.
Not sure how you would get around in Czech Republic without a rental car. There are buses and trains but I don't know how convenient and frequent they are. You could also look into a private driver.