You don't really need help, your analysis is correct.
"if I go first" is a subordinate clause, with "I" as subject and "go" as verb, "first" as adverb."
Try a more complex form with the same meaning:
Would you mind if I were to go first?
Would you mind if I went first?
In these cases, we'd have "go" in the subjunctive. There are only a few traces of the subjunctive in modern English, but it does crop up from time to time.