We made a shared Google Map to help you plan your time in town. It has much more on it than what we've included below, so we recommend checking it out and saving it. It's linked below.
On the University of Wisconsin's student union's terrace you can get food, drinks, and ice cream while sitting by the water. Although it's part of the University, pretty much everyone from the community goes here. A staple of summer in Madison.
It's free to go inside the Capitol and it's a pretty cool building that is well worth checking out. Make sure to go upstairs to the observation deck for a great view of the isthmus. If you're in town Saturday morning, there's a huge farmer's market on the streets surrounding the Capitol.
Fancy cheese shop by the Capitol where knowledgeable cheesemongers will teach you more about cheese than you ever wanted to know.
Nice little hike out to a peninsula in Lake Mendota. It's about a mile out to the tip and is completely flat so it's a great low effort, high payoff hike close to downtown. There are some additional trails around Picnic Point if you want to go farther.
Great spot to rent kayaks and paddleboards to get out on the water and see the city from a different angle.
Free modern art museum on State Street. The exhibits change often but are usually pretty good and the building itself is pretty and has great views of the area, so it's worth stopping in here if you are exploring the area.
The Great Dane is a Madison staple and a place we go quite often. The menu is pretty diverse so it has something for everyone, and if you're at the downtown Great Dane location there is a pretty nice outdoor biergarten where you can eat.
Iconic UW restaurant, like Triple XXX is to Purdue.
The coffee shop we live next door to. Pretty great coffee if you find yourself on the East Side of town.
Probably the best coffee shop on the Capitol square.