Are you going to lunch with Dan and I when do you use I and me?

The form 'Dan and I' is used as the subject of a sentence or a clause, and as subject complement after a linking verb (which restates the subject of the sentence).

The form 'Dan and me' is used as the object of a verb or a preposition, and as an object complement (which modifies or restates the direct object of the sentence).

Examples:

Are you going to lunch with Dan and me? (object of the preposition 'with')

Dan and I are going to lunch with you. (subject of the sentence)

They enjoyed the lunch that Dan and I brought. (subject of the relative clause)

They called Dan and me to go to lunch. (direct object of the verb 'called')

Their lunch-mates were Dan and I. (subject complement, restates the subject 'lunch-mates')

We were their lunch-mates, Dan and me. (object complement, restates the direct object 'lunch-mates')