The kind of story you describe--a children's story involving an animal which has a moral--is generally known as a fable. The most famous fables in the world are Aesop's Fables, and it is from them you can take your cue.
Fables are generally short, both because they are written for children and because they are written to make a concise, hard-to-miss point. There is not usually much character development because the characters are not usually human and they merely serve as an instrument to move the story along. There is nothing complex or complicated about the characters in a fable. The moral must be clearly seen in the story and clearly stated at the end of the story so that the children (and their parents) cannot miss the lesson you are trying to teach.
If the elements of a fable are simple, the introduction should probably not be complicated. A quick overview of Aesop's Fables reveals three kinds of beginnings.
- Once upon a time - this serves as a quick and effective signal that the story readers are about to experience is not particularly realistic and there is likely to be a lesson at the end.
- There once was - this serves the same purpose as "once upon a time" but offers a little variety. (Most English teachers do not especially like sentences that begin with "there," but it is an effective opening line for this style of writing.)
- Name the first place or character - just say something like "A chicken was very proud" or "An old man was very greedy." It is a fable, after all, and perhaps it is best just to get to the point.
If you are writing something which is longer and more complex, you should thinks of a way to capture the children's interest immediately. It may be that you have to write the rest of the story first in order to find the creative opening you want. Perhaps it will be a line one of your characters speaks throughout the story, or maybe it is a bit of foreshadowing about something which will come later. In any case, do not be afraid or unwilling to start the story just because you do not yet know what the first lines are. Start, and something will come to you. Happy writing!