The Dane County Regional Airport (4000 International Ln., 608/246-3391) has more than 85 nonstop flights daily.
City buses run to the airport. The closest bus stop is a couple of hundred yards after you walk left out the doors of the airport along International Lane.
A taxi ride from the airport to downtown costs $12 minimum.
Greyhound (800/231-2222) has lost its Madison terminal and had in one month moved to two different locations; at the time of writing it was opposite the Madison Metro North Transfer station on the north side.
Badger Bus (877/292-8259, www.badgerbus.com [1], $22) has seven daily departures to Milwaukee [2]: 3:30 a.m.–7:30 p.m., some via Mitchell International Airport [3]. Problem is: some leave from the UW Memorial Union, some from Kelley’s Market six blocks or so west of the State Capitol on West Washington Avenue, some from the Dutch Mill Park and Ride near the interstate. They have service to Minneapolis when the university is in session.
For Chicago head to the Memorial Union, where the Van Galder bus (608/257-8983 or 800/747-0994) leaves up to eight times daily: 2 a.m.–6 p.m. for O’Hare; some continue to downtown or Midway Airport. A one-way ticket downtown or to O’Hare is $27. The Van Galder also makes stops in Janesville and Beloit.
Links:
[1] http://www.badgerbus.com
[2] http://www.moon.com/destinations/wisconsin/milwaukee
[3] http://www.moon.com/destinations/wisconsin/milwaukee/sights/south-downtown/mitchell-international-airport