Madison has four trails that were combined into this 17 mile Capitol City Trail, which in turn connects to several other trails. This map shows the myriad of trails using colored lines (click to enlarge.) The trail is paved and in good repair.
I parked at Olin Park which can be reached off the Beltline Highway (US 12 and US 18), exiting at John Nolen Road and going north to Olin Park, just past the multi-building Alliant Energy Center which hosts expositions, conventions, concerts, and sporting events. The trail runs along Lake Monona with views of the downtown and Capitol building seen below...
Often along the trail you'll find helpful map kiosks like the one below, showing where you are, what's near you, and what other trails connect near you. The trail is well marked with signage so you don't miss a turn.
At Olin Park, you have the option of biking the Wingra Creek Trail which traverses park areas, crosses several streets, and gets you to the University of Wisconsin's Arboretum, the Henry Vilas Zoo and Park, which also offers more paved trails. and the dam which creates Lake Wingra as well as a public beach.
Heading south from Olin Park, you'll find 12 miles of trail, first briefly paralleling John Nolen Road, then passing beneath Nolen. The next section is run by the county, not Madison, and requires a DNR pass ($4 for a day pass or $20 for an annual pass) which can be purchased at a registration station. After paying, the trail deteriorates a bit, with frequent bumpy sections as you pass the huge sewage facility on both sides of the road you are alongside, and then the trail finally heads through rolling hillsides of meadows, then marshes, and later woodlands. You'll see mown grass pathways which are the E-Way, a cross country ski trail, and then the Lussier Family Heritage Center which has a wildlife viewing area, prairie restoration area, campground, washrooms, showers, water, and vending machines. By this point, I had biked 20+ miles in 95 degree heat, so I returned to my vehicle and missed the remaining 9 miles or so of what would have been the best natural scenery still ahead.