When pushing a child on a swing, you must give them a push periodically to keep them moving. this seems to go against the law of inertia and the conservation of energy. why does the child need a push periodically to keep moving.

Well if you know anything about physics then you should know that the force of gravity always acts on an object especially those that the force of tension is acting on therefore, the force you push the child with will only go so far until it's useless due to friction, gravity, etc.. Basically Newtons third law would be at play for every action their is an equal and opposite reaction. 

I Hope That Makes Senses :D