“He and I” is grammatically correct when it is placed before the verb and is the subject of the sentence.
“He and me ” is incorrect and should be “him and me” but only correct when “him and me” are the objects of the sentence.
He and I are going to the cinema.
The most important thing to remember is that “I” is the subject pronoun(the person that does the action) and “me” is the object pronoun(the person that receives the action.
When “He and me” is incorrect
“He and me” is incorrect at the beginning of the sentence because we do not use “me” as the subject of the sentence.
We are so used to subject(I) + verb that when this structure is disrupted, it can be easy to make a mistake.
I remember being corrected as a child that it is always “my friends and I “ and finding it difficult to say this structure.
I play tennis.
Subject verb object
She loves me
He and I play tennis
When to use “He and Me ”(Him and Me)
“Me” is an object pronoun that means that you receive the action. “Him” is the object pronoun of “he”
He punched me.
In this example, you received the punch. It doesn’t make sense to say “me punch”.
You can only use “him and me ” when you and the other person(he) receives the action.
They started a fight with him and me.
The artist painted a picture of him and me.
It is just a form of politeness to put “him” before “me”.