Oʻahu is the island people joke about: the one with the traffic, the high-rises and the tour buses. The jokes are half right: Honolulu is a modern American city on a volcanic island, and then there is everything else. Waikīkī was the first capital of the Kingdom of Hawaiʻi in 1795 and 1796, and downtown Honolulu still holds the only royal palace on US soil. Pearl Harbor, where the United States entered World War II, is a 30-minute drive from the beach.
The island works because it is small and has four very different coasts. The south shore is the city and the calm water. The east is lava cliffs and a drowned crater full of fish. The Windward side faces the trade winds, with mountain walls and soft sand. The North Shore is surf country, a different place in winter and summer. Five days sees all four if you plan by side of the island, and leaves a few hours a day for doing nothing on a beach, which is the point.
This plan is organized by geography. Day one is downtown Honolulu and Waikīkī, day two Pearl Harbor, day three the east shore with Diamond Head and Hanauma Bay, day four the Windward coast with Kailua, Lanikai, Kualoa and a luau, and day five the North Shore. Start on a Wednesday: ʻIolani Palace is closed Sunday and Monday, Hanauma Bay is closed Monday and Tuesday, and the Polynesian Cultural Center is usually closed Wednesday and Sunday, so this order lands each of them on a day it is open. Days four and five are long drives, from Waikīkī or from a base on that side. More on the country is on the United States page.