I and me: imagine you are the only one in the sentence:

Me and my friend went to see the new movie.

Erase my friend.

Me went to see the new movie.

So, "My friend and I went to see the new movie" is correct

That's an old picture of my friend and I!

Erase my friend.

That's an old picture of I!

So, "That's an old picture of me and my friend" is correct.