The city centre, Barbican, and Plymouth Hoe are all walkable from each other — the distances look larger on a map than they feel on foot. From the train station to the Barbican is about 15 minutes' walk; to Plymouth Hoe slightly less.
Citybus and First Devon & Cornwall run frequent bus services across the urban area. The ferry to Mount Batten (from the Barbican Pontoon) is the most scenic short journey; the Cremyll Ferry to Mount Edgcumbe runs from the Stonehouse area (Admiral's Hard). Both are passenger-only and run regularly in summer.
Parking is available in several central car parks but is expensive during daytime hours. If you're driving to the Barbican, the Coxside car park on the east side of Sutton Pool is the best option. Taxis and rideshare apps work throughout the city.