What kind of map should you carry on a day hike with children?
For me, the answer always is more than one type.
I like road maps to show me how to get to the trailhead and printed guides to give me an overview of the trail's key points. I like a satellite pic to show me where to park and how to reach the trailhead, as they are not always easy to visually locate. I use topo maps to show me the trail and landmarks on the horizon I can see to help orient myself. With a yellow highlighter I mark the trail and direction of travel on my topo map.
Fortunately, you don't need to spend any money on maps or mapping software. Road maps, topo maps and satellite pictures are all available online for free. If you're going to make a number of hikes in a specific area, such as a national forest or a national park, purchasing a large map of that entire region can be useful in selecting a trail to hike, though.
When carrying paper maps, you'll want to place them in a waterproof bag. You can buy see-through map cases that string around your neck, which I recommend for those walking in rainy climates. Since you're mainly going to walk on sunny days, however, a quart-size re-sealable plastic storage bag (like Ziploc) usually is fine.
During the hike, you'll want to check your topo map regularly to make sure you're still on the right trail. Doing so at rest stops or at any fork in the path is a good idea.
Have your children carry duplicate copies of paper maps in their backpacks. Should yours get lost or wet, they'll then have one. If you have older teens, they even may need to use them in case you are injured. Paper also makes great tinder if you end up spending the night in the wilds and must make a fire; burning a duplicate map won't be a major loss to the expedition.
And don't forget, if your children are old enough, teach them to use the maps by having them help you navigate the trail!
Read more about day hiking with children in my guidebook Hikes with Tykes.