If you're fearful of running into wild animals on the trail, rest easy as you probably won't. In fact, you'll probably see very little wildlife other than birds, squirrels, lizards and the occasional bug. If you do see a wild animal of any size - deer, fox, coyote - it certainly will be the highlight of your hike. Still, there's always the possibility that you'll stumble across a potentially dangerous animal.
A strategy of avoidance is best. Don't be tempted to feed wild animals. Besides inviting an aggressive attack on you, it teaches animals that humans are a source of food, meaning some are less likely to hunt or graze on their own but instead become a nuisance and even a threat to people. Such is the case with many national parks bears, who've learned to raid dumpsters and tents and even break into vehicles for food.
Also, don't pick up animals, as their natural inclination is to bite and claw when held. It's not cute to have a child get close to a wild animal for a picture when the creature bites or mauls them.
Related article:
g Statistical oddity: Animal attacks on day hikes
Read more about day hiking with children in my guidebook Hikes with Tykes.