Which Is Correct Your Father and I or Your Father and Me?

A very simple answer: If you are using as Subject, it will be: "My father and I", and if you are using as an Object, it will be: "My father and me".

Do you say my Dad and I or me and my Dad?

Use "I" when it is the subject of a sentence and use "me" when it is the object of a verb or preposition. Sometimes it can be easier to hear which one is correct if you simplify the sentence: Dad asked John and I/me to tidy the room. → Dad asked I/me to tidy the room.

What is the rule for using I or me?

Sometimes it can be tricky to determine if you should be using "me" or "I" in a sentence. Use the pronoun "I" when the person speaking is doing the action, either alone or with someone else. Use the pronoun "me" when the person speaking is receiving the action of the verb in some way, either directly or indirectly.