Maybe opinions are changing on this one, but the way I learned it is simple: refer to yourself as if the other person wasn't there. So:
Come to the store with my Dad and I. (no)
Come to the the store with I.
Come to the store with my Dad and me. (yes)
Come to the the store with me.
Dad and me went to the store. (no)
Me went to the store.
Dad and I went to the store. (yes)
I went to the store.
Remove the third party and it's immediately obvious which to use.
