I’ll second the recommendation NOT to go to Hanauma Bay.
Get out of Honolulu as often as you can, and see the North Shore.
Stop at the Pali Overlook north of Honolulu on your way to Kailua and its very nice beach (already mentioned).
My sister lives in Kailua, and I visit that beach a lot. Snorkeling at the beach is good, and if you have the stamina, try snorkeling out to the Mokes, a pair of islands about a quarter of a mile offshore (you’re only allowed to go on one – the other one is a bird sanctuary). If you don’t want to snorkel out there, rent a kayak to get you there, and snorkel with the island as your base.
If you’re really adventurous/brave/stupid, check out the cliff diving on the backside of the Moke (it’s horseshoe shaped). You can read about it in Oahu Revealed (already mentioned).
Check out the lesser traveled and lesser developed leeward side, on the west side of the island. It’s quite a contrast to the Waikiki scene. But read about it first, as it’s not to everyone’s liking. But you will see the “real Hawaii” there, if any of that exists anymore.
Skip the Dole Plantation in the center of the island unless you have an interest in agriculture. It’s pretty touristy, but you can actually learn a bit about how to grow pineapples.
On a rainy day, go to the Bishop Museum and stay all day. The Waikiki Aquarium is also nice.
Eat lots of pulled pork from any of the many Hawaiian BBQ joints, and be prepared to get a scoop of macaroni salad on the side. It’s very good, but you might tire of it because it’s a side dish that’s served with just about every restaurant meal anywhere in Hawaii! Try poi once, just to say you tried it.
Have a blast!