I hate to give you this kind of answer, but it depends so much on what kind of music you like.  Very tough to find live bands and good food in the same venue downtown.  You could try Buddy Guy's Legends if you like blues and southern/cajun food.  But it's kind of a tourist trap...and the food isn't exactly anything to write home about.  You'll have more (and better) options if you are willing to go get good food at one spot and take in a live band at another!  Oh...and if you are willing to take a cab/Uber out of the downtown area, so much the better!  Not sure when you will be in town, but check out the website for Dusek's / Thalia Hall in the Pilsen neighborhood.  It's very close to downtown.  Dusek's has great food and drink and is attached to the Thalia Hall venue, which brings in loads of cool music.

Feel free to pm me if you want additional suggestions

I have suggested before but City Winery on Randolph is always a good choice plus all the bars and restraints on Randolph are hard to beat and you can still walk to Greek town

Reggie Rock Club on state st. I go there all the time its amazing!!!!!!