Most of these activities are free, all are in walking/busing distance.
Olbrich Botanical Gardens
Minor League Baseball ($, Fri. 6:35)
Henry Vilas Zoo
UW Arboretum
Brittingham Boats ($)
Lake Monona Cruises ($$)
Dane County Farmer's Market (Sat. 6-2)
Picnic Point (nice walk to a lake view)
State Street
Madison also has lots of parks and biking (rentals, e-bike stations).
Most of these museums are free, all are in walking distance.
WI State Capitol Tours
MSN Museum of Contemporary Art
WI Historical Museum
WI Veterans Museum
MSN Children's Museum ($)
Chazen Museum of Art
UW Geology Museum
UW Physics Museum (closed on weekends)
If you've got a car, these are all within an hour drive.
National Mustard Museum (15m)
Taliesin East (50m)
Taliesin Preservation (50m)
House on the Rock (1h)
Cave of the Mounds (35m)
Devil's Lake (1h)
Wisconsin Dells (1h)
Ho-Chunk Casino (20m)