If youre going to rent a car be aware that the place you're staying at will charge an extra set amount per day to park your car on thie premises. That being said i've been to Honolulu these past 2 winters and never felt it necessary to rent a car. They have an excellent open air trolley system for getting around the city and n excellent transit system.
http://www.waikikitrolley.com/
http://www.thebus.org/Pop/beaches.asp
There is also some good shuttle services that will take you anywhere you want to go. We used the one below when in Honolulu.
http://viptrans.com/airport-and-waikiki-shuttle.asp
Sea Life park
http://www.sealifeparkhawaii.com/hawaii-parks/sea-life-park-description.asp
Hanauma Bay.
http://www.hawaiiweb.com/html/hanauma_bay_beach.html
Hiking to the top of Diamond Head is good as you get a nice view at the top.
http://www.lavasurfer.com/info/hikediamondhead.html
You might want to check out Atlantis Adventures.
http://res.atlantisadventures.com/tour_info.php?site=9&tour=59
There's also the flea market/swap meet at Aloha Stadium.
http://www.discover-oahu.com/aloha-flea-market.html
If you like water slides and such there's Hawaiian Waters Adventure.
http://www.hawaiianwaters.com/
There's also the inter island ferry if you choose to rent a car and want to see some of the other islands.
http://www.hawaiisuperferry.com/
If you want to see the whole island i would suggest taking a Circle Island tour. preferably with a company that uses small shuttle buses. The drivers are very knowledgale and will give you the history and a good description of each place you stop. It's also an all day tour. The conciere at your hotel will help you with this. Below is an example of such tour.
http://www.enoa.com/tours/majestic-grand-circle-island-tour.asp
The North Shore of Oahu is a must see also.
http://www.northshoreoahu.org/html/attractions.html