Which is correct: My Friend and Me or my friend and I?
That depends on where you and your friend are in the sentence. In colloquial speech me is often used where standard grammar requires I especially when someone else is mentioned too. Sometimes people use I instead of me because they know me is sometimes wrong, but have not understood the principle. (Others resort to myself which can sound rather pompous.)
• I am the subject of the sentence, but the object of the sentence is me.
If in doubt, take your friend out of the sentence.
• Me and my friend went to a party last night. [Wrong]
• I and my friend went to a party last night.
• My friend and me went to a party last night. [Wrong]
• My friend and I went to a party last night.
• The mayor has invited me and my husband.
• The mayor has invited I and my husband. [Wrong]
• The mayor has invited my husband and me.
• The mayor has invited my husband and I. [Wrong]
Incidentally, saying my friend and I instead of I and my friend is not better grammar. It’s just being polite.