Take the east entrance into the park (off of Zero). Follow the road, staying to the left (like you were going to the golf course). Before you get to the golf course club house, you'll find, on your left, a small parking lot with trail signage.
This paved trail provides a comfortable hike, with the exception of one large hill. You can even take a stroller or bikes. It is long (probably 2 miles one way), but of course, you can turn around at any point in the hike. You have a good chance of seeing some wildlife on this trail, including deer, and you’ll also see a pond.
If your kids are older and you want to dedicate the day to this adventure, consider connecting to the Janet Huckabee River Valley Nature Center trail after you finish this trail. When the Ben Geren trail ends on Massard Road, you’re nearly in view of the Nature Center. Just turn right (south) and you’ll be there in a few steps. Just remember that you’ll have to get back to your car.