For the most part, it is legal to ride on the sidewalk in and around Madison. More details are below. If you decide to ride on the sidewalk, be nice to people walking, and be aware that people in cars may not expect you at intersections or driveways.
Madison
Riding your bike on the sidewalk is generally allowed in Madison. The exception is where buildings extend all the way to sidewalk, as is the case in many parts of downtown or in commercial districts such as Williamson Street or Monroe Street. The City has a brochure giving you the details. Another exception are the inside sidewalks on the Capitol Square. No biking allowed there, because the state government says so. If you ever need the exact city ordinance:
(1) No person shall ride a bicycle on the sidewalk where a building abuts the sidewalk. Bicycle riding on sidewalks is permitted, except as prohibited in this subsection and otherwise regulated in this chapter.
12.76 SPECIAL RULES APPLICABLE TO BICYCLES.
Monona
Riding your bike on the sidewalk is generally allowed in Monona. The exception is Monona Drive between Winnequah and the city limits with Madison:
(E) Operation on Sidewalks.
(1)Bicycles shall not be operated on Monona Drive sidewalks from the northern limits of the City to Winnequah Road and on the Yahara Cove Boardwalk at River Place.
(2)Whenever any person is riding a bicycle upon a sidewalk, such person shall yield the right-of-way to any pedestrian and shall give audible signal before overtaking and passing such pedestrian.
City Ordinance