Megabus runs several times per day from Minneapolis directly to downtown Madison.
Badger Bus runs regularly between Milwaukee and Madison and drops off right downtown. Fares are reasonable.
Van Galder bus lines frequently run to Madison from Midway, O’Hare, and downtown Chicago. They are a simple and quick way to get to downtown Madison if you’re coming from Chicago, or flying in there.
Van Galder Website
Madison is very bike-friendly, with many bike paths, bike lanes, and shared-road streets.
Madison has an easy-to-use bus system that runs pretty much wherever you want to go. Trips are $2. If you have an Android phone, the Busradar Madison Metro App gives accurate route and schedule information.
Madison Metro Ride Guide (contains complete schedule, literally everything)
Weekday Single Page Map
UW Campus Single Page Map