It marks you as ignorant of the English language, although practically nobody will call you on it these days. Him and me are object pronouns that are incorrectly used as subject pronouns. The corrected sentence should read, “He and I went to the movies.”
The simplest way to remember whether a subject pronoun (I, he, she, it, we, you, and they) or an object pronoun (me, him, her, us, you, and them) is correct is to ask: Are the pronouns doing the action? Or are they the recipients of an action done by others? In the above sentence, He and I are doing the action of going to the movies; consequently, subject pronouns are necessary. In this sentence, “John gave him and me tickets to the movies,” object pronouns should be used because him and me are recipients of John’s generosity.
Other aspects of pronoun usage can be more complicated, but this is a good place to start.