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's simply courteous as well as grammatically correct.
Secondly, when speaking of yourself and another person, there are times to use "me" and times to use "I".
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.)
He and I drove down to the beach. (... ... I drove down to the beach.)
Look at this photo of him and me. (Look at this photo of ... ... me.)
We ate lunch and then he and I left. (We ate lunch and then .. ... I 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).
My spouse is clamoring for breakfast. I'll go make oatmeal for him and me. Later, after lunch, I'll lie down for a nap.