TheBus is Honolulu's public bus transportation around the island. From the airport, take a # 19 or 20 to Waikiki. The last bus to Waikiki leaves the airport at 11:45pm on weekdays and at 11:15pm on weekends. Buses run about 20 minutes apart during the day and 30 minutes apart in the evenings, and the ride duration is about 1 hour and 10 minutes.
The fare is $2 one way, and a transfer is issued at time the cash fare is paid, which is good for 2 hours. It is good for one use and must be handed in when boarding.
There's also a 4-day pass for visitors for $20, which is available at ABC stores in Waikiki, the Ala Moana Shopping Center, and at the Bus Pass office (811 Middle St., Honolulu, HI 96819), which is open Monday to Friday 7:30am to 3:30pm, excluding holidays.
Popular sites and attractions are available through this good public transportaion system.
Pickup a system wide Oahu bus map to familiarize yourself of the system.