(Boy is this off topic!)
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.
Regardless, 'me' is always correct as the object of a prepositional phrase because it's an object pronoun. 'I' is a subject pronoun.
(Boy is this off topic!)