Trains in Italy - Naples to Sorrento

Ciao tmurphy — Giangilberto from Milan here. Good news: you haven't missed anything, and you're actually all set. The Rome–Naples ticket was the only part you can buy in advance, because the Naples–Sorrento leg works like a metro, not like a mainline train.
Here's how it goes on the day. Your train arrives at Napoli Centrale. Inside the station, follow the signs downstairs to "Circumvesuviana" — the station underneath Centrale is called Napoli Garibaldi. You buy the ticket right there at the windows or machines (a few euros), and that's completely normal: the Circumvesuviana has no reservations and no advance sales, you just take the next departure. Trains run roughly every half hour and it's about 70 minutes to Sorrento, which is the last stop, so you can't miss it. Fair warning: it's a commuter line — perfectly functional, but crowded, no luggage racks, stops everywhere. Keep your bag in front of you and you're fine. Small bonus: it stops at Pompei Scavi and Ercolano, handy if those day trips are on your list.
The Campania Express you found is exactly the answer to your instinct: same route, same platforms, but the tourist version — guaranteed seat, air conditioning, room for suitcases, only a handful of stops. It runs seasonally from spring through October, a few departures a day, costs a bit more, and this one you can buy online in advance on the EAV website (EAV is the company that runs both trains). Traveling with luggage in summer, it's worth every cent.
Two alternatives worth knowing about. In season there are fast ferries from Naples' port (Molo Beverello — a short taxi or Alibus ride from the station) straight to Sorrento: about 45 minutes, and arriving by sea with Vesuvius behind you is the best entrance the town has. And if you happen to land at Naples airport rather than arriving by train, the Curreri bus runs direct to Sorrento without touching the city at all.
Are you traveling with big suitcases, and which month? That changes which option I'd push you toward.

Tmurphy,
There is some great info here. Would you mind expanding your title to include Sorrento? That would greatly help others doing a Search. This is a frequent question.
Safe travels!

Would you mind expanding your title to include Sorrento? That would greatly help others doing a Search.

That isn't necessary - the search engine extends to the text of each post - a search on "train sorrento" included this post in the result set.

Thanks tmurphy!