Car / Taxi / Bus, Oahu
The bus is the only real transportation. They don't have a metro... or many taxis outside of honolulu. So I would suggest renting a car or your vacation is gonna SUCK. The bus is slow and doesn't come often or on schedule. On the north shore it comes about every half hour. It takes like 2 hours to get from the north shore back into town. (double the time)
Use "The Bus" for getting around in Waikiki and the Island of Oahu, as driving your own car can end up being a hasle due to trafic and hard to find and expensive parking.
There are frequent routes going allmost anywhere and the cost is $2 per ride, but make sure to ask the driver for a transfer and you can catch another bus within two hours.
A four day pass is available for $20, and may be convenient if you plan to get on and off the bus a lot.
For getting around Honolulu and Waikiki The Bus is the best option, and it takes 1 1/2 - 2 hours to get to the North Shore from Waikiki.
An Oahu Island roundtrip takes around 4 hours.
Rent a Jeep . . . I was worried about being able to rent a jeep because some of the chain rental agencies said that they were sold out. I did pre-book my jeep before I got to the Island but I did find out that there are many Jeep/car rental places around the hotels that have a wide variety of jeeps/cars available.
Rent a car. The island is easy to navigate and has good roads all around the island. Unless you want to sit on Waikiki the whole time a car is the way to go.