It's been decades since I studied grade school grammar and I'll admit that my own grammar is not always correct. However, the misuse of pronouns when talking about "him and me" is rampant. There is a simple rule that makes it easy to determine whether to use "I" or "me".
First of all, when speaking of another person and yourself always mention the other person first, i.e., "him and me", or "he and I". Never "I and he", or "me and him". It is proper (and polite) to refer to the other person first just as it is polite to allow the other person to walk through a door first. It is simple courtesy as well as grammatically correct.
Secondly, when speaking of yourself and another person, there are times to use "me" (the objective pronoun) and times to use "I" (the subjective pronoun).
Here's an easy test to determine when to use "I" and when to use "me". Simply remove "he and" or "him and" from your sentence and you will automatically know whether to use "I" or "me". It's easy as pie.
My mother gave him and me a sandwich. (My mother gave ... ... me a sandwich.) objective of the verb "gave"
He and I drove down to the beach. (... ... I drove down to the beach.) subject to the verb "drove"
Look at this photo of him and me. (Look at this photo of ... ... me.) object of the preposition "of"
We ate lunch and then he and I left. (We ate lunch and then .. ... I left.) subject to the verb "left"
Please don't get me started on the verbs "lie" and "lay". For the proper use of "lie, lay, lain" or "lay, laid, laid" go to this website. It's easier to give you a link than to try to explain it myself. (Said with a wink and a smile).
It's bedtime. My hubby and I (subject) should be in bed. He (subject) is already asleep. I'm not.
P.S. Only my most faithful followers will realize that this is a reprint from early 2008 but it is still valid today. Grammar has not changed. He and I (subject to the verb) should always recognize that it's good to know that grammar rules also apply to him and me. (object of the preposition "to")
Funny what a good high school English teacher can drum into your head. All that underlining and sentence diagramming paid off. Thank you, Mrs. Vanderlinden. You patiently put up with a lot of flack from our small class of 20.