I: subject; me: predicate. When talking of two people: reduce it to one. "Me and her went away" is reduced to "Me went away. " Sounds wrong--therefore I is correct.
"I" is the person who is doing something. (I go. I run. I feel. ) "Me" is the person to whom something is done. (She likes me. He hits me. ) "Me" is also used after a preposition, which is a word like to, from, at, in, with. (They go with me. You give it to me. )