There are two competing companies running trains from Naples to Rome (and to Florence, Venice etc.): Trenitalia and www.Italotreno.it/en
Take a Freccia if you want, but Italotreno's fares are usually cheaper than Trenitalia's and their T&C are more flexible. The complimentary on-board wifi is easier to use.
There is a manned luggage deposit at Napoli Centrale, but security takes time.
Don't be paranoid, but Be aware of your surroundings while walking around Naples. Remember that pickpockets need crowds to do their "job" and go away without being noticed.