This is not so much an English grammar rule as it is English grammar etiquette that says something about the speaker. When referring to yourself and anyone else, the polite thing to do is to put yourself last, not first.
Example: They took up a collection for me and him. (wrong)
Example: They took up a collection for him and me. (correct)
Example: John gave all of his baseball cards to me, Bill, and Bill’s brother. (wrong)
Example: John gave all of his baseball cards to Bill, Bill’s brother, and me. (correct)
Example: The professor wants to talk to me and you after class. (wrong)
Example: The professor wants to talk to you and me after class. (correct)
There’s been a great deal of talk in recent years about the descent of common courtesy into rudeness and the change of beneficence to self-absorption. Using the “me and anyone else” construction instead of the “anyone else and me” construction is just another sad sign of this cultural reality, and it makes a personal statement about the speaker. It’s up to you to decide how you want to present yourself.