First of all, Cynthia, grammer is correctly spelled grammar. As for your daughter using it correctly, I’m not sure it depends on where you live or where you grew up. It depends more on a commitment to writing and speaking well.
To answer your original question, the proper order for this sentence is: Amber, Courtney, and I went to the movies.
You need to replace Me with I because you need a subject pronoun here and not an object pronoun.