Honolulu International Airport is about 25 minute's drive west of Waikiki via Ala Moana Blvd/Hwy 92 (Nimitz Hwy) or the H-1. You can also catch a ride between the airport and Waikiki on a public bus (about an hour) a shuttle bus (45 minutes) or a taxi. Many of the larger resort hotels offer free shuttles to their guests.
TheBus is Honolulu's public bus network. Its routes branch across the island with each line's destination written above the bus' windshield. The Ala Moana Center is the central transfer point. Overall the buses are in excellent condition - clean and air-conditioned - though buses on popular routes tend to be packed and their pace is always dawdling. Setting your watch by this system gives you nothing but a good sense of Hawaiian Time. The Waikiki Trolley is an expensive tourist-laden open-air bus geared primarily for sightseeing shopaholics. The attraction-lined route between Waikiki's Royal Hawaiian Shopping Center and downtown Honolulu is narrated.
Oahu is not a big island and few places are more than an hour's drive from Honolulu. If you plan on spending all your time in the resorts of Waikiki forget about renting but if you plan to get beyond the city limits a car is the easiest way to do it. The minimum age to drive in Hawaii is 18 years and most car rental agencies hike that limit to 25. Gasoline is about 25% more expensive on the island than on the US mainland. Driving is on the right.
Taxis wait at most major downtown hotels and at the airport. Otherwise you'll need to phone for a cab. Bikes are available for rent in Honolulu and Waikiki and most bike shops provide maps helmets and locks. The city is poorly suited for cycling though and most riders prefer to use their bikes for longer jaunts around Oahu.