Rome to Positano

The most convenient way is to have someone drive you there. But if you'd rather rely on public transport: Train from Rome to Naples (70 min) http://www.trenitalia.com Commuter Circumvesuviana train from Naples to Sorrento (70 min) http://www.vesuviana.com Bus from Sorrento to Positano (50 min)
http://www.sitasudtrasporti.it/public/tpl/campania/5070.pdf

My wife and I just returned from Rome & Positano. If I remember correctly, the trip via train/bus would have entailed 2 separate trains, and 1 or 2 bus transfers. The only way that made sense to us was driving. It took us less than 3 hrs and as long as you're not there during a holiday period, when the highways & secondary roads will be packed, it was a pleasant trip. The train/bus method looked impractical, time consuming and if you're hauling some luggage - potentially a real pain.

You DO have to consider what you'll do with the car while in Positano - park it, turn it in (is there somewhere to do that in Positano?), drive around the area?...so if you don't want the train (personally, I had no problems with it), then be sure and do some research into how a car would work for you.

If you rent a car, there are several pay lots in Positano. Parking in Positano is not cheap, up to 25 euro a day, although if you stay over 2 days you might save some money and get away with 15-18 euro a day. Hourly parking is close to 3-4 euro per hour. Although expensive, however if your stay isn't that long, you might find it convenient to have a car to get around because buses aren't frequent and practically not available at all in the evening if you want to have dinner in another town (such as Praiano or Amalfi). But if all you want to do is stay put in Positano and maybe take a trip to Amalfi or to Capri (by boat), then a car isn't needed much, you can rely on the bus. Spending $100-120 day for rental cost+parking a day to keep the car parked there, may not be worthwhile in that case. If you intend to return the rental car upon arrival, after you check in in your hotel and drop off your luggage, you need to drive back to Sorrento and return your car (no car rental companies in Positano). Once you return your car in Sorrento, you can hop on a Sita bus and go back to your hotel in Positano (50 min bus ride).