Badger Prairie County Park is a quick 15-20 minute drive from Madison and offers nice trails to walk the dogs through prairie areas. It also has an off-leash area as well.
Note: The Yelp site map seems to be pinpointing the park on the wrong side of Hwy 151. If you're coming from Madison you should exit Hwy 151 in Verona at Business 151 and take a quick right. You then will turn into a drive that appears like you're going to the Badger Prairie Health Center but just go past that and you'll enter the park.
We first checked out the "mountain biking" trail head that departs from the first parking lot, next to the off-leash area. This trail goes through a wee bit of prairie and then into woods briefly. I wasn't wild about this trail because it borders a rather busy road so you hear a bit of traffic as you're walking which I find counter-intuitive to what I'm looking for. There is a second parking lot, about 1/2 a mile past the first, and this seems like a better point to start from. The prairie is nicer here, longer grass, more wildflowers, less traffic nearby.
The off-leash dog park is sizable and well-maintained. It was moderately busy at 6pm on a Tuesday; you can see the trails and grassy areas in the posted photos.
All in all, another great option. I probably like the Lake Farm park a bit better if I had to choose between the two simply because it seems more "in the middle of nowhere". However, Badger Prairie has a number of paved trails lacing through it that seemed to be popular with inline skaters and bikers. There also is another access point off of McKee Rd that I will need to investigate!