I've done all three but not in one day. Of them all climbing Vesuvius was the least enjoyable, it's not really a climb, a bus takes you most of the way and then it's an easy walk up a narrow trail with hundreds of other tourists. The crater is not what you'd imagine (or certainly not what I imagined when I visited pre-internet days).
Took the train, no problems with pick pockets (neither has anyone else I've know who've taken it).
Pompeii is spectacular, huge and very crowded. Herculaneum is much smaller, much quieter and actually allows you to take in more than Pompeii does so in that respect I prefer Herculaneum. The site is a short downhill walk from Ercolano Scavi railway station.
If you only have one day then I would see Herculaneum first then visit Pompeii and stay as long as time permits. Skip climbing Mt. Vesuvius. It can be see from all around the area and it doesn't need to be climbed to appreciate its presence. Of course, if time allows all three to be done then by all means fit in Vesuvius if it's something you want to do.