The most popular and most economical route is by train. The Termini Train station is in the center of the city and tickets can be bought there on the same day of travel, beforehand, online or if you have a RailEurope pass you can reserve your ticket months beforehand (A RailEurope pass is also known as a Eurail pass).
The cost of your ticket depends on whether you choose first or second class, but tickets usually range from 17 to 51 Euros or $26 to $76. First class seats are usually bigger, more comfortable material, more spacious legroom and less people per compartment. Do not buy a second class ticket and sit in first class as you will be fined if the train inspector catches you (they check tickets on every journey). The journey by train lasts 1 ½ to 2 ½ hours.
The cost of a flight will obviously depend upon the day and month you want to travel and whether or not you want a one-way or return fare, but Alitalia has flights that start from as low as 48 Euros or $72. However, remember you will still have costs associated with getting to and from the airport. A plane generally only takes 55 minutes, however don't forget time spent in customs, and check-in lines.
If you do not want to take public transportation with other people and don't feel like driving yourself you can rent a taxi. A taxi from Rome to Naples will cost you approximately 218 Euros or $325 as of October, 2009. A taxi ride takes about 2 ½ hours.
