Introduce no more than two characters at a time. One character is best, if possible, but two is acceptable if the story calls for it. Begin with the main character in a children's story and introduce the character by the name you will use throughout the book. A line like "Jack Rabbit woke up withe the sun shining through his window" is an example of an ideal opening. Establish the character's name within the first paragraph, preferably within one or two sentences. This tells the child who they are reading about as quickly as possible.
Set up the character's main personality as soon as possible, reinforcing your character's name. An example following the name of the character might be, "Jack Rabbit was a shy bunny who had trouble making friends." This introduces something about the character and reinforces the name of the character.
Introduce the next character in the same way as the first, after having first established your initial character. You may reinforce your initial character's name somewhere in the second character introduction if the second character is closely related to your first. Here's an example: "Mary Rabbit was Jack Rabbit's only friend, who happened to be his sister too. Mary Rabbit was more friendly than Jack Rabbit, so she would always try to introduce him to her friends." While it is preferable to stay with the name you gave your character in the beginning, you may call Jack rabbit simply "Jack" at this point, since his character name is already established.
Introduce additional characters with the same system, as the story requires. Name first, then something about the personality of the character. After each of your character's have been introduced, find a pace that keeps them in the story. Always use the full name you started with when a character has been off the page for any length of time, just to refresh your readers about who the character is.