Both 'me' and 'I' are pronouns, but that doesn't mean they are interchangeable. In fact, by the rules of English grammar there are specific times to use each one. So how do you know when to use which of these two pronouns?
In technical terms, as explained by the Oxford Dictionary, you should use 'I' when the pronoun is the subject of a verb and you should use 'me' when it is the verb's object.
Okay, great, that's the technical explanation, but when you're in the middle of a crowded bar trying to prove your correction-friendly acquaintance wrong, how do you explain it in a way other people will understand? Try this trick: remove the other people/subjects from the sentence. Let's look at a couple examples.
"My dog and me ate ice cream yesterday." Remove "dog" and what are you left with? "Me ate ice cream yesterday." It's pretty clear that's wrong! So this is an instance where you would need to change 'me' to 'I.' Let's look at another one:
"My mother joined my father and I for dinner." Remove "father" and what do you have? "My mother joined I for dinner." Nope. Make that a 'me!'
Use this simple deletion and substitution trick, and you'll find yourself becoming a pronoun usage expert in conversation and in writing.