O'ahu:
I say rent a car and take a trip around the island. This is one of my favorite things to do. Drive from Hawaii Kai towards Kaneohe right around the island up to the North Shore and stop when the road stops. Many tourists do not get to see this but it is highly recommended. There are many places to pull over, take pics and then jump into the water.
A word on swimming and jumping in the water: if you don't see any local people in the water and only toursits, it's a pretty good indication that there is a reason for that. Be warned that the undertow at Hawai'i beaches can be pretty strong. Watch out for Portuguese Man O' War jellyfish... they're sting-y.
For Hawaiian food, I recommend Ono Hawaiian Food in Kapahulu. It's a hole in the wall but it's wonderful. There is also Hawaiian Food in Ala Moana shopping center in one of the ugliest food courts ever created. That place is called The Poi Bowl. (And if you want good Chinese food, go to Patti's Chinese Kitchen in the same ugly food court.) Try poi at least once, don't be a poi hater.
Spend a couple of hours in Waikiki and that's it. It's gross, touristy and lame. If you go to the Cheesecake Factory there, you may see a waitress that looks like me. It's my sis.
If you want a good plate lunch go to any one of the L&L Drive-Ins that you encounter. But for an awesome plate lunch, go to Rainbow Drive in in Kapahulu.
If you must, go to the Polynesian Cultural Center. It's wonderful. Although I myself do not support that place with my money, it's a cool romp. The evening show is quite good.
Get shave ice from Matsumoto's in Haleiwa. It's good. If the line's too long, go to Aoki's... it's pretty much the same there.
For hiking, I recommend the Diamond Head trail, Makapu'u trail and a trail that takes you into Koko Head Crater.
I don't know a whole lot about the shows and luau-type events, but there are a number of them. If you want just a mellow time with good Hawaiian music, go to Chai's in the Aloha Tower Marketplace. They have live performances while you eat. Robert Cazimero is there on Tuesdays, I believe. He's great.
If I think of more, I'll let you know.