It is an easy enough walk (about 1 mile) from the port to the Circumvesuviana station in Via Garibaldi, Naples. Take train (cheap return ticket) to either Pompei or Ercolano. Ercolano is nearer (20 mins), the ruins there are a smaller version of Pompei and easily explored inside about 2 hours (so less walking and sunburn!). A trip (minibus, I think) to Vesuvius can be booked at modest cost from an office adjacent to Ercolano station - I ddn't take it so have no further details.
Why not get a taxi to the Naples train station (about €10) and then get a train to Pompei (this costs next to nothing). The volvano is very close to this. The train is signed destination Sorrento and that place is absolute magic. You can even do a round trip and come back from Sorrento on a boat that comes very close to the ship.
Beware the Naples taxi drivers, they can be a pain in the bum. They follow and hassle you to take a tour with them, it may be good, it may not be, but I didnt feel all that safe.
As Jones said the Naples taxi men are a pain but having been there many times, let them approach you, tell them you are going to Vesuvius by train. Keep walking & yes they will follow you. Refuse the price, keep walking, one by one they will get lower & lower in price until you get the price you think is fair. My experience with is, although they feel threatening whilst negotiating, once thats over with they have been very sociable after that & we've had some good days with them & they do stick to the deal. Last year we paid 120 euro for 5 of us, round trip, in a taxi van to Sorrento as we were on a short stop in Naples, & the ferry times didn't work out too well. He sat & waited for about 3 hours or so & returned us to the ship. Sorrento is 50 km, Vesuvius is about 20 km so that should give you a price guide. Alternatively if your feeling a bit more ambitious, look at the train option.