"My friend and I...." 'Proper' grammar, or 'Correct' grammar?

Okay, so I know that 'My friend and I' is PROPER, but does that mean that 'Me and my friend' is incorrect or just improper? Proper, is similar to capitalizing names, but if you don't capitalize the names, then it is technically 'improper' not 'incorrect'. So is using the phrase 'Me and my friend' incorrect? Because it offers no incorrect grammar in my opinion. Your opinion?