It's very easy. Take the train from Rome to Salerno. Right in front of the train station, board a SITA bus headed toward Amalfi. You'll need to purchase a ticket first (only 1-2 euros) from a tabacci stand or the tourist info office in the station. Vietri Sul Mare will be the first town you come to once you get out of Salerno. The ride takes only a few minutes. There are also taxis in front of the train station - of course you'd want to ask a driver how much the ride would cost, but I would guess perhaps 20 euros.
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