We often hear teachers avoiding the obviously erroneous "Him and me went to the store," without having the slightest idea of why it is wrong, substituting "He and I" for every "him and me" construction, whether correct or not, putting them into the wonderful category we call the "graduates of the Nathan Detroit school of grammar," (aka. the jailhouse learned; use of pretentiously incorrect grammar is as telling as that of the use of speech peppered with incorrect or incorrectly used words of more than two syllables)

The correct use of these is as follows:
* If the pronouns are the subject of the verb, use the subject form (he, she, it, we, I, they):
o I went to the store. He went to the store.
+ He and I went to the store.

o She went to the store. I went to the store.
+ She and I went to the store.
* If the pronouns are the object of the verb, use the object form (him, her, it, us, me, them):
o He gave some candy to us.
o He gave some candy to her. He gave some candy to me.
+ He gave some candy to her and me.
o He gave some candy to him. He gave some candy to me.
+ He gave some candy to him and me.

* The rule of thumb here is to test out the sentence in your head with only one pronoun. You wouldn't say, "They gave something to he," would you? You wouldn't say, "They gave something to I," would you? No, of course not. Then why would you say, "They gave something to he and I?" It only demonstrates a wish to speak correctly without the knowledge to do so (if you hear a teacher speaking like this, consider it a warning of a lack of knowledge but a wish for it).

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