Yes, the market still runs, even in winter! They move it indoors to the Monona Terrace, which is conveniently attached to the Hilton! Well worth a visit.
Two out-of-town groomsmen (Nick and Adam) highly recommend!
If you need to hang near the airport for a bit and enjoy a brew and good bar food, this is the place for that! (If you need other local beer recommendations, feel free to talk to the groom--this is Wisconsin, after all.)
The groom once worked in a state Capitol (in North Dakota) and loves taking first-time Madison visitors here. It's the center of the city and will be ever-present on wedding day.
Free art within walking distance of the hotel!
UW's art museum, also free and situated in a pair of buildings that are an architectural landmark on campus.