You can leave from Naples from three different ports to arrive in Procida.
Napoli Beverello - the closest, just 5 minutes by taxi from the station.
Napoli Mergellina - you can get there by train or metro, less traffic than Beverello.
Pozzouli - you can get there on the metro. Interesting town, with lots of ROman ruins and volcanic activity.
Check out times with CAREMAR, several other companies as well, but they have most service and best website.
For long distance and with luggage, Microtaxis are a great alternative. Also, they can navigate the narrow streets much better than the normal taxis and you are not sandwiched like on the bus (which at night it is easy to miss your stop!)