Answer:
If it would be "I" alone, then it is "Joe and I".
Example 1: "I am going out", "Joe and I are going out" (subject of sentence).
If it would be "me" alone, then it is "Joe and me".
Example 2: "She gave it to me", "She gave it to Joe and me" (object of sentence).
There is no need ever to get it wrong if you follow this simple rule.