Indian Lake County Park is a solid option for various outdoor activities if you are in the Madison area.
It has a range of hiking trails, from easy to fairly strenuous, and you can tailor a hike to suit any time you have available. The park is dog friendly if your dog is leashed (there is a use fee if you bring a dog), and there is actually a dog park on the west side of the park. The dog park is near the boat launch, which is a solid launch for paddling or any other boating activities. The lake is shallow and usually choked with algae and weeds as you get into the summer. It's okay for a quick paddle to work out the kinks, but there are far better paddles near Madison.
They are redoing it right now, but there is a shelter and fire pit that my outdoor group likes to use for bonfires in the fall and spring. Hopefully that work will be done soon.