Which one is correct:
I and my father are going to the market.
Me and my father are going to the market.
The second is not grammatical.
The first is grammatical (except that it lacks the verb 'are'), but neither idiomatic, nor conforming to speech etiquette. Politeness requires that one mentions oneself second. So one would normally say:
My father and I are going to the market.
In determining whether to say 'he and I', or 'he and me', just mentally omit the 'he'. Clearly one would not say me am going to the market, so it has to be 'I'. But one would say It belongs to my father and me, since 'I' am no longer the subject of the sentence, and the accusative pronoun is called for.