Who/Whom or Whoever/Whomever??

The correct form is "Whom are you going out with?"

"Who" is used when the word is meant as a subject; "whom" is used when the word is meant as an object. In cases where the sentence is a question, one helpful way to look at it is to rewrite the sentence as a statement. For example, your sentence, "Who/whom are you going out with?", if rewritten as a statement, would look like this: "You are going out with who/whom." When it is written as a statement rather than a question, it is easier to see that the "who/whom" is an object rather than a subject (because the subject of the sentence is clearly "you"). Hence, the sentence should take the objective form: "whom."