The best trail closest to you is Ft. Ben State Park. Some are paved, some are stone, the park is great. You can push the stroller around on the paved trails easy. Basicaly old roads that are well maintained. I have been taking my daughter there for day hikes year round, for many years. We go there on snow days too. But technically you are not to have weapons in the park. I always do.
Anything goes in National Forest lands, like the Hoosier National.
A lot of State Forest land is cool too, like Martin State Forest. The key is many of these areas permit hunting so firearms are not a big issue. Martin has a cool nature loop set up for kids. Worth the drive if you want to get out of town. Just West of Shoals, IN.
If you want more info call me on the cell, but with a littel one, Ft. Ben's paved trails are good.