Can't help you too much on the bar scene. Like most German cities, Braunschweig has an Altstadt zone which has plenty of pubs & cafes, but unfortunately I can't think of any specific ones to recommend.

If you have a free day or two I would recommend visiting Goslar, which is about 1 hr south of Braunschweig by train. This is a smallish town in the Harz foothills with all the original 14th-15th century buildings still intact. I love going there.