"Whom" is correct according to formal grammar, but it is very uncommon these days. And, according to formal grammar, it actually should be "With whom" at the beginning of the question. (Some grammar purists would argue that one should not end a sentence with a preposition.) In the U.S., we rarely use "whom." It is much more common to use "who," even in academic writing.

Technically, it should be "with whom are you going?" But, you're unlikely to find anyone who actually talks that way. You're more likely to hear "Who are you going with?" or "Who ya gonna go with?" or "Who ya goin' with?"