The paved 1/4 mile trail at Rotary Park is smooth, has only a few other walkers and has the benefit of close parking, water fountains, picnic tables, restrooms, a playground, large open grassy area for free play and spray parks for water play (spray grounds are closed on Mondays for maintenance). This is a great place to walk with kids - you can see them from every part of the trail and there is enough variety to keep multiple children's interest while you walk/run. There is also a creek just off of the park if you enjoy exploring and letting your kids play there.
The paved trails at Country Club park are very similar - same 1/4 mile smooth trail, intermittent shade to walk in. Country Club park/trail has more shade with large Oak trees and a nice breeze usually moving across the North part of the trail. The play area is perfect for younger children; the small steps and little slides are easy to navigate. Older children get bored very quickly on this playground unless you bring activities, but the large grassy area and basketball court are perfect for sports. Children can also explore the creek or bring a blanket for a picnic. There are no restrooms, no water fountains, no picnic tables at this park/trail, but the pretty view, quiet and shade are a nice place to walk.
*As with most parks in Independence, the majority of people/walkers are very nice and friendly, however parks are also popular for people who are vagrant and sometimes for those buying/selling drugs. I have never had a problem at these trails but practicing usual precautions is wise as at any park/trail.