First Avenue and the 7th St. Entry are an institution (they're in the same building so that technically isn't a grammatical screw-up). I'm not sure if any bands are playing there during this period that you'd necessarily be interested in (Wiz Khalifa? Gene Ween? Cave Singers?) but it's at least worth a visit. So's the Triple Rock Social Club, which Damian from F'ed Up stands by, and for good reason. Even if there's not a band playing there that you'd like, they've got a great bar and food menu. Incidentally, if you go barhopping and you're a beer nerd, it's your civic duty to order a Surly. It's Minneapolis' most notable microbrew and they put out some great, hoppy beers.
Also: if you have even the remotest interest in baseball, Target Field is absolutely 100% worth a visit, and this week includes homestands against the White Sox and Padres. Not sure how easy it is to get tickets, but it's worth a shot.
Seconding the record store namedrops, the Walker and the Bryant-Lake Bowl.