Mount Vesuvius is perhaps best known for its eruption in Roman times (24 August 79 AD), described by Pliny the younger, when Pompeii and Herculaneum were destroyed. The eruption left a large crater which has grown and shrunk with subsequent eruptions and this can still be visited today. It is a currently dormant volcano that occasionally emits streams of lava (the last eruption was in 1944; although its dormant state could be an indication of a build-up of pressure and a coming explosive eruption). At the top of the volcano is a crater rim that affords a view into the crater that still fumes slightly. Besides that you will have (only on clear days, but the mountain is notoriously covered in fog or clouds) a stunning panorama overseeing the Bay of Naples, Naples, Capri, Ischia, the edge of the Amalfi Coast and more.
Be warned: Going there will make you sweat a lot and set you back about €15 (including a private taxi up the hill).
By public transport
The quickest / cheapest way up is to take the Circumvesuviana railway (Sorrento line) to Ercolano (Scavi) then follow the Ercolano instructions, or Pompeii, where Vesuvio Express may also run from, Ercolano is closer though. You can buy tickets on the lower level of the Central Station aka Stazione Centrale a Piazza Garibaldi. Also as another option, you could bus to Ercolano Scavi station in Herculaneum or the Pompei Scavi station, but this will probably be slower.
Alternatively, there's a direct bus-line that leaves twice a day from Naples Central Station square and goes straight up to the visitor parking. It's run by one of the minor bus companies (not the city transport) and you might have a bit of a hard time finding it, but unless you have a pre-arranged group tour it's perfect. It runs on a timetable, so it's less capricious than the taxi service from Ercolano. Two-way ticket from Naples Station costs €13.50. Bus leaves at 09:20 and 10:35 at the Garibaldi station.
At Ercolano, Vesuvio Express can take you the rest of the way, their office is on the left as you exit the station. They offer a minibus ride up and down the mountain for €10 each person. For €20, they will offer a ticket with entry to the crater. Note: They are able to get special entry tickets for cheap, so when you purchase the package, they keep more than the €10. However, if you buy your own ticket to the crater, it's still €10 so you may as well pay it to them and skip the queue. They will typically wait until a minibus is full and then leave. They will wait for all of you at the upper car park for 90 minutes, which is generally about right. From the car park you'll have to hike up the last bit before you can actually see anything exciting. They say the walk is about 800m long (optimistic), going uphill most of the time and ending at the 1170m mark. Wear good walking shoes as the trail is covered with dust and rock pebbles.
If you like sticks and having something heavy to carry, don't forget to borrow a stick off one of the guys offering them on the trail, on your way down, don't forgot to "tip for the stick"!
Also, remember to tip your driver, you could have plummeted if he hadn't honked at every corner, or have gotten there much more slowly if he hadn't sped past that bus on that cliff edge.
There is word on the Internet of a local bus and a handful of other options for getting up the volcano, however the tourist information office advises that the only method is Vesuvio Express. The tourist information office is closed on Sundays.
You can also do a 6km hike to the summit along the winding roads. To do this you will need to take the local bus service ANM 5 or 176 from Ercolano to San Vitto. From there you will be able to walk to the summit. You can take in the views of the national park and enjoy the local wildlife. You will still need to pay the entrance fee when you arrive at the crater, but for the energetic ones it is a must!
There is a bus running from Pompeii Scavi to Vesuvius. The ticket is €10. The bus does not operate during the winter season.
A small company organizes mini-bus trip to Vesuvius at the exit of the Pompeii Scavi train station. The ticket costs €20. The company claims to wait one and half hour at the top, but the driver may only allow one hour, which is very tight. It is advised to ask the organizer to confirm one and half hours duration in front of the bus driver.
You can get to the upper parking area, but be prepared to part with at least €2.50 for the parking fee. And it's a good idea to go there earlier in the morning to avoid the jams that may happen along the narrow road.
There are a couple of huts selling cheap and sometimes bizarrely inappropriate souvenirs and expensive drinks around the rim. Take a bottle of water with you to avoid these excessive charges.
Weather: even on hot days you can get strong and chilly winds at the rim so take a lightweight fleece to wear at the top.
Everyone has to walk steeply uphill the last 860m. That includes bus loads of tourists which, on busy days, will congest the path and give up halfway. Closed-in, strong shoes are essential, as the path is dust and loose rock. You might want to get hold of one or two walking sticks provided. And there is a small entrance fee of €10 (adults), €8 (students).
By mountain bike
That's Amore, the leader cycling company, arrange on demand daily excursion along rough authorized paths of Vesuvius volcano opened for this company by the park. Can be possible ride short, easy or strong routes.
Eat & Drink
Vesuvius is an active volcano and may erupt any time in the near future. Over the last few centuries, Vesuvius has erupted at intervals ranging from 18 months to 7½ years, making the current lull the longest in 500 years. Its lack of eruption may merely indicate a build-up of pressure, which may result in a more explosive next eruption, posing a lethal hazard to over 500,000 residents living in the same place that got destroyed in 79AD.
Listen to local authorities if there is any news about the state of Vesuvius, and do not wander any further than the trails go.