Rapaire, if you are going through books and maps, you will find more places of interest than you could visit in a month of years.

Try and catch a concert (free) by The Royal Hawaiian Band and soloists. Schedule at
Concert Calendar
The band has a long history, started in the 19th C. by the King. It is currently an agency of the City and County of Honolulu. They frequently played (Sunday) in the bandshell in the park to the east of Waikiki; I think they vary locations more now.

Like Clontarf83, I have enjoyed people watching at the beach in Waikiki. A good place is by the chess tables (newspaper vendors, and other "working class" players, who take the game seriously).
Toward the east end of Waikiki Beach is Hilton Hawaiian Village and Hale Koa, near Fort DeRussy. The beach is not crowded here, and in the mornings, instruction in Tai Chi is given near the "Fort" in the recreation area (open to the public). Don't know schedule, but a call to the Fort should tell. Hale Koa is a first class resort for active and retired service personnel.
The Hilton (22 acres) has many facilities for non-guests. Lapperts Ice Cream in the varied shopping area, an outdoor snack bar which had excellent inexpensive mahi-mahi sandwiches and fresh fruit platters, overlooking the beach; When in Honolulu I would eat lunch there on occasion.

Look for a hotel or inn in Honolulu with kitchen facilities or at least a fridge and coffee machine (some are very reasonable). Flights from W. Canada often arrived near midnight. I would take a hotel room (previously reserved) at one of the cheaper 2-3 star (AAA) places near the main Waikiki drag, check in, and go to the all-night grocery plus (several) and get local fruit, Hawaiian Portuguese bread (a round loaf, don't miss!), Hawaiian dark honey and milk for breakfast in the room (and a six-pack of Dos Equis).
Saturday nights, the kids come to town from the sticks, and drive the main drag in their four-wheelers, etc. Stroll along, watch the patrolling cops chatting up the working girls.

In your rental take the Tantalus road above Honolulu. Great views, and upscale residential with fine plantings. Also a good art museum.

In the city, take a look at Punahou School, where the well-to-do send their sons, including Barack Obama (His grandmother was a VP of Bank of Hawaii, and stepfather Soetero was independantly wealthy- so much for the food stamp nonsense). Digression- his mother's (Dunham) Ph.D thesis is finally receiving full publication; scholars in the field consider it a classic.