rome to naples, by the fastest trains, takes about 1h45mn, there are 2-3trains an hour though not all are non-stop. to get to sorrento you must change stations in naples - napoli centrale to napoli collegamento fs (adjacent to centrale) and take the circumvesuviana ferrovia local train to sorrento, journey 55mn. I have no fares information, you should contact someone like trains europe - http://www.trainseurope.co.uk/
cheapest option is to take a diretto train from roma termini to napoli centrale then walk downstairs at napoli centrale .For local train to sorrento it costs aprrox 5 euros to sorrento im going by dec 2005 prices when i was last there . The current price for a diretto single in 10.12 euros between roma termini and napoli centrale if you wanted the faster intercity plus service (i c plus) it would cost you 17.53 euros single these are the current prices obtained from trenitalia check out there website at www.trenitalia.it/en/index.html to obtain fares put in stations and times then click on standard in fares category box to obtain price .
forgot to say journey times are on trenitalia website. also if you do take this trip dont forget to validate your ticket at the yellow machines on train stations otherwise you will be in line for a hefty fine from the conductor on the train hope you enjoy this trip