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)
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.