Remember These Rules
The easiest way to figure out which form is correct is to remove the other person from the equation. For example,
"Joe and me went to the party."
"Joe and I went to the party."
Well, by removing Joe from the sentence, you'll have all you need to solve this dilemma. This is what you are left with:
"Me went to the party" or "I went to the party."
It is easy to see which is correct—"Joe and I went to the party."
Let's try another one:
"Martha gave Joe and me our paychecks."
"Martha gave Joe and I our paychecks."
Again, remove Joe from the sentences, and you are left with:
"Martha gave me our paychecks" or "Martha gave I our paychecks."
Obviously, "Martha gave Joe and me our paychecks" is the proper form.
What about "myself?" Well, it should never be used as a substitute for the word "me," but is generally reserved for emphasis. For example, you would not say, "Tell myself what's on your mind," or "Come to myself with your problems." You might, however, say, "I'm scared of dogs, myself."