Circumvesuviana

No reservations - it's just a little local commuter train with zero frills. Just buy tickets when you get there.

It is possible you will stand for the first several stops. I think there is a bus you can take instead, might be more comfortable.

This is the only official site: http://www.eavsrl.it/web/circumvesuviana but tim is right, it's like a city bus on rails

In season you could also use the Campania Express train whose tickets can be purchased on-line and that has only reserved seats.

The Circumvesuviana is a bit "colourful" and often a haven for pickpockets and scammers. You might be more comfortable on the Campania Express (assuming it's running at that time) - http://www.napoliunplugged.com/campania-express-tourist-train-to-herculaneum-pompeii-and-sorrento

As stated earlier, there are no reservations on the Circumvesuviana. Just buy tickets in the Garibaldi station and board the train. I've never experienced any problems in the few times I've ridden it, but there have been posts here from people who have had problems. Watch your luggage carefully during the trip and be sure to wear Money Belts.

If someone has not guided you to this site as an excellent resource for train travel in Europe, I suggest http://seat61.com . Seat61 has a British orientation, but the info is good for most travelers. Looks like you've got some good experienced advice in any case.

You'll be taking the Circumvesuviana as well if traveling from Sorrento to Pompeii: very inexpensive and takes about 30 minutes or so. It's sort of bare-bones thing but gets the job done! The Express is a nice alternative but has far fewer daily runs so isn't always convenient.

Current schedule from Sorrento to Naples:
http://www.eavsrl.it/web/sites/default/files/eavferro/Sorrento%20-%20Napoli_0.pdf

Naples to Sorrento:
http://www.eavsrl.it/web/sites/default/files/eavferro/Napoli%20-%20Sorrento_1.pd

There is a direct bus from the airport to Sorrento--destination is across the street from the train station in Sorrento. That's what I should suggest--can't remember it's website, however, it's a walk-up and get on system.

I understand that there is also a bus directly from the Naples Airport to Sorrento, which has been mentioned on this forum many times, although I have not taken the bus and do not know the details of schedule, cost, etc. It has always sounded like a good deal, if it works for you. Maybe someone else will provide more info about the bus.
To take the train, you would need to go from the airport to the Naples train station, and then to the train. Is something of a hassle, and that and the train itself might be more of a hassle assuming you have baggage. The Circumvesuviana train is really like an old city bus, it can be very crowded esp. during commuter rush hours, and it has something of a reputation, so hold your stuff closely and in a secure way.
One of the better splurges in Italy, IMHO, is to take a taxi or car service from Naples direct to your lodging in Sorrento. Costs about 100 euros, but is so much easier and quicker than making connections and buying various tickets or shorter taxi rides.

Jerry, it's the Curreri Viaggi http://www.curreriviaggi.it/

With our bags, we will take a private car from Napoli Centrale train station to Sorrento, which costs 95 Euro, but on the way back it's the Curreri bus to Napoli Capodichino airport for 10 Euro.

I rationalize it by averaging the cost!

We took the bus from Naples Airport to Sorrento in May -- €10 each, air conditioned, bags placed below the bus by the driver. Ride was scenic, somewhat reminiscent of the Amalfi coast bus. Good deal!!

I missed the part that you'd be arriving at the airport, and would also recommend the Curreri Viaggi Bus for that trip.