Potawatomi State Park is the easiest nature break from Sturgeon Bay. It covers wooded shoreline on the bay side of the peninsula, with trails, water views, and the eastern terminus of the Ice Age National Scenic Trail.
The park is close enough to use for a short walk, but it also rewards a slower visit if you want to get out of town without driving deep into Door County. In autumn, the mix of forest and water is especially good.