Define a child's moral development at different stages?

There is really no such thing, if you wanted a conservative answer. Psychologically speaking, this answer has been addressed by Kohlberg's (stages of moral development) but whether these stages are true is open to lots of debate. Its rather outdated, this sort of theory, really. Philosophically speaking, you can't really ask about stages of moral development without defining what morality is in the first place. What follows is really how children develop to follow rules, from Kohlberg's theory, which is the best that you will get:

Level 1 (Pre-Conventional) : :: 1. Obedience and punishment orientation ::: :::: (How can I avoid punishment?) :: 2. Self-interest orientation ::: :::: (What's in it for me?) : Level 2 (Conventional) : :: ::: 3. Interpersonal accord and conformity :::: ::::: (The good boy/good girl attitude) ::: 4. Authority and social-order maintaining orientation :::: ::::: (Law and order morality) : Level 3 (Post-Conventional) : :: ::: 5. Social contract orientation ::: 6. Universal ethical principles :::: ::::: (Principled conscience) (source: wikipedia)