When we landed at Naples airport we had to catch the Alibus to Piazza Municipio or Molo Beverello, the Port of Naples. It can get a little hairy so here's what you do to avoid stress. Outside of the Arrivals doorway across the street is the bus stop. Go stand under the sign for Alibus. You'll see orange ANM buses as well, these are the local buses. We took the Alibus which is clearly marked for 3 euro apiece. You can buy the tickets on the bus so don't worry about it. The Alibus only has two stops, at Piazza Garibaldi and Piazza Municipio. You can't miss Piazza Municipio there's a HUGE castle there, Castel Nuovo. Get off the bus, walk down the sidewalk to the left of the castle, cross the street and you are at Molo Beverello. There will be a red and white snack stand to your left and ticket offices to your right. Choose whatever mode of transportation you want to take and then walk to the left towards the snack bar. Turn to the right and you'll see an LED display that tells you when your boat will arrive and at what dock. Seems simple but it's stressful if you're trying to figure it out as you go!
When you get to Ischia Porto walk straight until you hit the sidewalk and then turn to the right. Go to the Tabacchi which is behind the gelato store to buy your bus tickets (you'll see the sign for the Tabacchi but all you'll see is the gelato display), for however many days you need them (I think they have one, two, three and seven days). Come out of the Tabacchi, turn right and head up the hill. You can't miss the multitude of buses that will be across the street in a turnaround and that's where you catch the bus. Happy traveling!
The buses in Ischia aren't that complicated. The CS and CD buses circle the island continually and stop at all the main towns. The CD bus has many fewer stops, it takes the CS bus twice as long to make the same route. One bus circles the island to the right and one bus circles the island to the left. The only part that gets complicated is that the bus stops have numbers only, not names and bus drivers are not God's most helpful people. Sometimes you would see the sign for Forio or Lacco Ameno or Cassamicciola Terme and you would be passing out of the town, have to get off at the next stop and walk backwards! I was able to get a schedule for the buses from a friendly news vendor but the schedule does not have maps. The buses do run however from 6:30 in the morning until midnight and on a very regular basis.
You can also purchase biglietti on the bus but you'll get what they call a "regional" ticket that's not good for very long, I was never clear on how long and they cost 1.50 euro. Since we only used them when we wanted to go one way that was fine with us. However if you are staying on the island for a while it is possible to buy a bus ticket good for a number of days at a discount.
The buses were our main mode of transport and other than one breaking down we never had any problems finding or catching a bus once we figured out where the stop was in Panza!
These high speed ferries - called hydrofoils - make quick work out of transporting passengers to and from Ischia. It is a comfortable 50 minute ride to / from Naples in the wide and comfortable seats.
Remember to pay for your luggage tax (1 or 2 Euros per suitcase) before you enter the hydrofoil. (If you forget, someone will track you down and collect it)
We took the hydrofoil from Molo Beverello. Took about 45 minutes to get to Ischia Porto and cost 18 euro.