Since the stadium is way out in the middle of nowhere, we are just driving up for the game, will have a drink and some food in the car. We are planning on hitting the bars afterward (in case you didn't know, Buffalo bars close at 4 am, so a lot of times, people show up at bars around 11 pm). A lot of young people (18-22) do go out to bars on Chippewa Street downtown. Those who are 30+ avoid it like wearing masks against the plague. People around 25-35 hang out in Allentown & Elmwood (bars like Anchor, Founding Fathers, the Old Pink, I'd recommend Founding Fathers), while 30-45s congregate around Hertel Ave (Moor Room, etc.). If you're at a hotel outside of town, the only two suburbs that have a lively nightlife are Main Street in Williamsville and town center of East Aurora (Bar-Bill Tavern).
There are a lot of good breweries in downtown Buffalo if you're into beer, like Big Ditch and Thin Man. My favorite is Community Beer Works, but it's in a neighborhood on the West Side. A good bet to get good food and drink is Dinosaur BBQ if you're into BBQ.
If you want to try buffalo wings downtown, I'd recommend Gabriel's Gate instead of Anchor (the place that originated them), and if you're in the burbs, look for a Duff's or else Bar-Bill in either East Aurora or Williamsville. Most people I know think Gabriel's Gate are the best, with Duff's and Bar-Bill a close #2.