"I" is a subject, as in "I went to the store." "Me" is an object, as in "give it to me."
The problem develops from the beating we all took in grade school about making our pronouns agree, as in "He and I went to school," instead of "Him and me went to school." So, what's the problem?
The correct usuage would be, "give it to him and me," and "between him and me." The easiest way to understand the correct usage is to remove the first pronoun. You wouldn't say, "give it to I" or "she spoke to I."
Don't be afraid to use "me" and "him" correctly. Their misuse is like fingernails on a chalkboard to anyone who knows even a little grammar.
Speaking well is part of good leadership, good business and, frankly, part of good social etiquette.
@wesbleed Book Wes to speak at your next event.