We're great together, you and I. For it is only I that accompanies you. But alas there come times you and I must part and you should instead be together with me. It's not always easy to see, but sometimes you really should be with me.
No one appreciates you and I. They only appreciate you and me. It matters not if they use your name. For I gets left out just the same.
You and I are a compound subject. But when you are with me, we make a compound object, see? For with or without you, I am the subject, but the object is always me.

Just remember that whether we're alone or not, it is I that does things and that things are always done to me.
Grammar Girl does a good job explaining this in her
Between You and Me post. Other references can also be found by performing an internet search using grammar, compound object, and You and I as keywords.
Come August 4th, Myra and I will have been married for twenty-eight years! (I am now bowing to your awesome applause.) But twenty-eight years ago, wedding gifts were given to Myra and me.
No, I am not an authority on grammar. I do, however, try to abide by the rules of the language when I know them. I break them intentionally at times. I break them without intending to sometimes as well. As is the case with most of us, I am still learning and will continue to learn. Let us improve together.