Between islands: Inter-island flights are the standard way to hop between islands. Hawaiian Airlines and Southwest Airlines operate frequent short flights -- most are under 45 minutes. Book in advance during peak season as fares rise quickly. There is no longer a regular inter-island ferry service, though small boat operators run between some islands.
On Oahu: Honolulu has TheBus, an extensive public bus system that covers most of the island for a flat fare. It is reliable and inexpensive. The Skyline rail system connects western Oahu to the airport and downtown. Waikiki is walkable. For the North Shore or windward side, a rental car is more convenient.
On the other islands: You will almost certainly want a rental car on Maui, the Big Island, and Kauai. Public transit on the neighbor islands is limited and infrequent. Roads are generally good but often two-lane, and traffic around resort areas can be slow. On Molokai and Lanai, rental options are very limited -- book well ahead.
Driving tips: Hawaii drives on the right. Speed limits are low and enforced. Be cautious on winding coastal roads, especially the road to Hana on Maui and the Na Pali Coast area on Kauai. Many rental agreements prohibit certain unpaved roads.