To begin your script you will need an idea. Many movies usually rely on an animal theme like "Ice Age," "Madagascar," and "Kung Fu Panda." There are also really successful children movies with human characters like "Polar Express" or "Monster House." Decide what you want to focus on and flesh out your story into a multi-page outline.
You should also envision your story as a live action, animated, or mixed media movie. If you figure this out early then it could help visualize your script and allow you to expand more if it is animated. Then it's easier to have talking animals or things like monsters.
Once the actual script writing has begun, you really need to put yourself into a children’s mind and see what they enjoy. If you some children of your own, or access to some, play off some of your ideas, or even act them out to see if the children will enjoy them.
Also study other forms of children’s entertainment from the Disney catalog, Nickelodeon, and other TV and movie production companies. Shows are popular for a reason and while you shouldn’t attempt to just duplicate something popular, adapt it into a unique creation of your own.
Adults have to sit through these movies too, so help keep them entertained with some concepts that may only be humorous to adults. Do not make it too risky though, because then you may end up offending parents instead of entertaining them.
Keep the plot of the movie pretty formulaic. You can throw in a few twists here and there, but the plot should remain pretty straightforward so that children can easily follow it.
A character arc and message is important in movies, like "Shrek" shows that beauty is in the eye of the beholder. Show a change in the character and deliver an uplifting message that will keep the children happy and optimistic.
The most important thing to work on is making the movie as fun as possible without sidetracking from the plot too much. Keep a lot of sight gags in the story, make the characters interesting and always keep the children in mind when you are writing.