As it is a small, quaint town, it is easiest to get around Portsmouth by car. Unlike a more metropolitan city which can easily be navigated on foot or by public transportation, Portsmouth is far too rural and suburban to rely on these types of transportation.
The Rhode Island Public Transit Authority RIPTA does, however, have busses running to and from Portsmouth. For the schedule go to: (http://www.ripta.com/)
Taking a cab is another great way to get around Portsmouth or the entire island is you don't have a car or if there isn't a bus route. There are multiple cab services on the island, such as Rainbow Cab Company, Cozy Cab, A 1 Car, Cab, and Van, and Orange Cab.
Rainbow Cab Company: 401.849.1333
Cozy Cab: 401-846-2500
A 1 Car, Cab, and Van: 401-841-0411
Orange Cab: 401-841-0030