If you are museum buff interested also in architecture, you should not miss passing by Brno and visit the Church of St. James, Spilberk Castle (which overlooks the city), church of the Holy Cross, the Cathedral of St. Peter and Paul (Petrov Cathedral Hill), Technical Museum and Schwartz palace. Aside from these, be sure to pass by the city's farmer's market (cabbage market) to buy homemade candies and preserves.
I visited Brno as a day trip and this place seemed friendlier, more laid-back and more fun than the capital Prague. You should check out Fleda (Stefanikova 24) and Duck Bar for live music events. There is also cool wine bars and pubs like Charlie's Hat Music Pub and The Pub, which has pour-your-own beer tap's on tables. All of these are located underground.
The Cathedral of St. Peter and Paul is quite something....specially the towers and crypt. Not far from the cathedral it's the Capuchin Church which is also worth passing by as it's home to a crypt filled with dozens and dozens of mummies.
If you are into wine, there are many villages around Brno where you can do wine tasting and wine tours around vineyards