The nearest International Airport is Naples Capodichino (code NAP). Once you have arrived, you'll have to take one of the methods listed below to cover the final leg of the journey.
From the airport you can take a bus for €3 (called Alibus) to Stazione Centrale in Naples (get off here for connections to Salerno via train). It also stops on Piazza Municipio near the ferry port, from which you can take a 5-minute walk to catch the Bus (SITA). The route is not obvious, so see the section below.
You can buy your ticket on the Alibus, and you can get to it by walking right out of the airport terminal to the bus stop which is less than 20 metres from the airport terminal exit.
From Naples Airport there is a SITA bus running directly to Salerno (Euros 7). This bus runs only 4 times a day, and not on Sunday. You have to walk some 300 meters to reach the bus
From Napoli Central or Piazza Garibaldi you can easily reach Salerno by train. The regional trains ticket to Salerno costs around €3-€4 and it's called U-5 Unico Campania. Other trains are more expensive and cost around 7-10 euros.
The traffic in Naples follows unwritten rules that are not accessible to foreigners (or even some Italians). Red traffic lights don't always stop the traffic (the locals know the 'important' traffic lights, and will stop at these). Mopeds regularly leave the road and drive on any other surface they can find. Follow the locals, and you should be OK.
You should ask the way to the "SITA" (the name of the bus company that provides the service). If you stand where the bus stops, and face the sea (with the tall castle to your right), you will need to cross the main road and head towards the left. After about 400 meters or so, you will see a small congregation of buses to your right. Check the destination written on the bus, or you can ask for "Salerno".
The service is quite rapid for the first part of the journey, but then will leave the motorway and take a more tortuous route. The journey should take an hour so so. The ticket should cost between €3 and €4 per person.