Vilas is a great park with ample green space and an excellent lake for swimming in the summer and ice skating in the winter. While pets may not be allowed in the park, you can certainly walk along the road/path with your pet.
Vilas beach offers a great swimming spot and a sand beach. Lake Wingra is a wake-free lake so it's less busy, quieter and presumably cleaner than other lakes around. There is a lifeguard stationed at the beach and a maintained swimming area. Note - be sure to check the City website to make sure the beach is open before heading out, Madison beaches can close unexpectedly due to bacteria (translation -don't feed the geese! They poop & then beaches end up closed!)
Vilas also offers a great playground for kids and ice skating in the winter. The only thing on my "wish list" for Vilas is that they would actually pave a sidewalk along the road through the park. You have to walk either on the grass or along the road, which can feel dangerous. It's also a one-way drive through the park but with some regularity drivers get confused and/or impatient and drive the wrong way through there so keep your eyes open for that!