Capital Springs/Lake Farm is nice, and very close since it's right off the Capital City Trail. They're still in the process of merging the state and county parks. It's dog accessible, so there's probably some way to finangle things around the elderly Lab.

New Glarus Woods is a tougher trip, but should mostly be accessible via state trails (Sugar River, Cap City and Badger State).

Governer Dodge is further yet, but it's on Military Ridge so it ought to be doable as the kids get older.

I keep looking at the state trail map and wishing that they'd finish linking Madison and Milwaukee and La Crosse. That'd make a lot more state parks accessible via bike, and it'd allow some more unconventional touring.