It isn't really a question of being old-fashioned; it is more based on how formal the training is. If you are enrolled in a course (e.g., at a school or university) then you are considered both a "student" and "pupil." But if you are studying something on your own, or through a more informal training course, then we only use "student."

You still hear it, but it is slowly fading out in usage. Your friend is correct in that, but the word is not so old-fashioned that anyone would be surprised if you used it. In that way, it is not like using "thee" or "thou" as "you" - that would definitely get you some surprised looks.