We stayed at the Sorrento Hilton and got told we couldn't visit Vesuvius by train from our hotel. It wasn't possible and we would need to take a paid-for coach excursion incorporating Pompeii (which we'd already seen). Some people like pre-packaged tours, but Italy has such fantastic public transport that you should consider using it. This guide has some hints and tips for getting there.
I recommend taking 1.5 litres of water and a warm jumper/rainproof top. Stout walking boots should be worn (not flip flops as the pumice hurts if trapped under foot).
Catch the train to Pompeii Scavi (excavations). From Sorrento at either the Tabac or the ticket platform:
Uno andata e ritorno per Pompeii per favore oono andatta ay raytourno pair pompay pair favooray.A return ticket should cost about 3 Euros from Sorrento. Observe the next departing train from Sorrento to Naples above the ticket booths in Sorrento station. Go to this platform. Catch a train. The journey will take about 30 minutes. Trains always depart on time. From the station, avoid the hustlers selling overpriced guides to everything (7 euros) - the official guides from the entrance to the excavations are much better (dark red colour).
At Pompeii train station, ask for a bus ticket for Vesuvius:
Uno andata e ritorno per Vesuvius per favore oono andatta ay raytourno pair vesuvius pair favooray.This will cost about 8 Euros 70 and covers the 30-60 minute journey to the 1000 m point.
Turn right out of the station, towards the masses of water sellers/fruit sellers. You should be heading downhill and you will go past Pompeii entrance. After 200 metres from the station, you will arrive at a crossroads and the entrance to the motorway (Autostrada).
Wait at the bus stop just outside the tourist information point:
You should be able to see the following:
After (up to) an hour, the bus will turn up. Buses always leave on time. Stamp your ticket in an orange machine near the driver. The driver will set off and head down the Autostrada towards Ercalano (optional on some journies) and/or Vesuvias. The bus may or may not travel through Ercalano and you can wonder why they still haven't patched up the falling down balconies on the tower blocks.
Scarey journey to the top of Vesuvius, past mounds of rubbish thrown out of cars/coaches. Note the use of the two-tone horn around bends. You will stop for 10-15 minutes about 200 metres from the top at the old chair lift station.
At the old chair lift station, in a true scooby-doo style, a ginger woman will appear and explain some of the history of Vesuvius and want to stamp your ticket. She will tell you about her dad who scared tourists before her. Don't worry, she's harmless and the ticket stamping is optional (IMHO). She will tell you there is no water/food/toilets at the top. This is not true. She is gutted that they haven't replaced the chair lift from which her family made their living. She has a bee in her bonnet about Italian Greenpeace and I think she has a point. And she would have got away with it too if it wasn't for those meddling kids. Note the semi-wild cats living around the cabin.
Walk uphill. Buy a national park ticket to keep the barriers from falling down. These cost 7 euros 60. You will be informed by large signs that you need a qualified vulcanologist to accompany you to the top. They have to make a living somehow, but we didn't bother. The walk is about an hour for the physically fit if you continue as far as you can.
Turn up at least 10 minutes before the bus is due to depart. The bus departs from the same place you were dropped off. The bus will leave on time or possibly early so be warned. The bus will stop back at the Tourist information office in Pompeii.
Don't listen to the robots at the Hilton.
Total cost: 3 Euros (train), 8 Euros 70 (bus), 7 Euros 60 (vesuvius nat. park)= 19 Euros 40