"He and she" is used as the subject of a verb.
Examples:
* He and she went to the store together.
* He and she are best friends.
"Him and her" is used as the object of a verb or preposition.
Examples:
* I saw him and her at the park.
* The teacher gave him and her a homework assignment.
Here's a simple way to remember:
* If you can replace "he and she" with "they," it's correct.
* If you can replace "him and her" with "them," it's correct.