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.
There appears to be a few trails crisscrossing the slope of Mt Vesuvius, according to the signboard near the ticket office. There's a trailhead that starts just behind the ticket office that doesn't seem to require an entrance.
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.
The direct bus-line to Vesuvius by EAV no longer runs from Naples Central Station square.
The EAV bus is cheaper but not available directly from Ercolano. You'd have to find a way to Torre del Greco and catch it from there. So, Vesuvio Express (8:30 — 15:00) is the main option going up. 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. Last bus departs at 15:00 daily. For €20, they will offer a ticket with entry to the crater. They will typically wait until a minibus is full and then leave. They will expect your stay at the crater to be 90 minutes, but as long as there are empty seats on any bus going down you can take any descending bus. They didn't even check the tickets actually. From the car park you'll have to hike up the last bit before you can actually see anything exciting. the walk is about 800m long 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.
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!
The cheapest way is to use an EAV air conditioned bus, for €3.10 each way. It can get quite full, so you may want to onboard it on its first stop on Piazza Amfiteatro. Timetables for departures and return are posted to the right of the ticket window in the Pompeii Scavi train station and are also available at the EAV website. The bus starts its journey from Piazza Amfiteatro (GPS: 40.74910, 14.49530), which is in front of the eastern entrance to the Pompeii ruins, this is the closest stop to the Pompeii train station. The next bus stop is in front of the Pompeii Scavi (Villa dei Misteri) train station (GPS: 40.74879, 14.48126), which is in front of the western entrance to the Pompeii ruins. Other tour operators might say the EAV doesn't go all the way up, but they are liars trying to get your money.
A small company organizes mini-bus trip to Vesuvius at the exit of the Pompeii Scavi train station. The return ticket costs €20, but out of peak season you could negotiate to €15. In normal traffic one-way takes 40 minutes. The company claims to wait one and half hour at the top, but if he is late 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 buy Vesuvius visitor tickets in the bus for the regular price of €10.
You can get to the upper parking area, but be prepared to part with at least €2.50 for the parking fee (in July 2016 the parking charge was €5). 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 is a minibus taxi service from the car park charge point to the ticket office for €2 per person, return.
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 an 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.