One of the first hikes we took with both girls was to the Audubon Society in the Northwest situated between Forest Park and Washington Park. Along with having an informational center and a bird sanctuary, behind the center is 4.5 miles of trails which all connect next to a small pond. Although I did see a couple of people with strollers, I do not consider this to be a stroller friendly hike as there are a few places with steps and large branches, not to mention some steep slopes. But for small children this is a great area of trails on well worn paths with some beautiful scenery.
It took us about 45 minutes to walk around the park with Beatix in a carrier and Violet walking. There is one short section which runs along Cornell Road and allows you to connect to trails in Forest Park if you want to extend your hike. (Here is a trail map.)