If you like animals, the Kahala resort in Hawaii Kai has dolphins in the back area. Just Park at the Wai Alae beach park on Kahala Ave (the resort is at the end of the street) and walk down the beach to the resort. They have dolphins and a sting ray, and a nice beach area to swim in.
Also, the Sheraton in Waikiki has a nice pool and hot tub. Enjoyed that many times and only got kicked out a few :)
The Hilton Hawaiian Village is on the corner of Ena and Ala Moana. They have many shops that you can walk around and visit and they have an area with penguins that you can stop and take pictures. There are also turtles in the pond with the penguins and you can reach down and pick some of them up. They will swim to you if you put your hands in the water. They love people!
Nearby on the corner of Kapiolani and Kaheka is one of my favorite restaurants, Shokudo Japanese restaurant. It's amazing. They have sushi pizza and a dish that looked like ginger shavings that moved by itself. Whatever it was, it was delicious!
While you are there, you should check out the Ala Moana shopping mall and get a smoothie at Shirokiya on the third floor by Macy's. They also have some find desserts and other food from Japan, and a departments store full of Japanese "as seen on tv" products.
Nearby in the mall, is Panya, another fantastic restaurant, that has a wonderful assortment of baked goods, as well as a full bar, Asian fusion menu and borsch, of all randomness. View their menu at http://panyagroup.com/default.asp
Genki Sushi, which I think of as the McDonald's of sushi, is located next door to Panya in the mall and has other locations on the island as well. It is an interesting experience for first timers, as you can sit at a table and snatch desired sushi plates as they make their way around the room on a conveyor belt. Not the best sushi, but fun, nonetheless.
For very delicious sushi, you can't do better than Sansei Sushi, located in the Marriot in Waikiki. On Sundays, they have sushi half off, which is well worth standing in line for (and you will). Call ahead to check times so you don't miss it.
Coffee or Tea? is the best coffee shop on the island by far and the only place to go for authentic bubble tea. It is located in McCully Shopping Center at Kapiolani and McCully. They also have chocolate milk with pearls or a delicious avacado smoothie with pearls if you are caffeine sensitive.
Not if, but when you go to the North Shore, stop and have shrimp at one of the shrimp trucks, such as Mackey's. If you take the H1 to the west side where it ends and travel up Farrington Hwy you can find the mostly deserted "Pray for Sex" (Makua) Beach, one of the places you can get away with some nude sunbathing and go rock diving. Or you can mix it up and have nude rock diving.
Find time on the way North to go to Lanikea Beach to see the sea turtles sunbathing on the shore. The biologists policing the beach will tell you not to get too close, but if they are not there, be warned there is a mortgage-sized fine for touching them.
Or go East through Hawaii Kai to Haunauma Bay to snorkle with the fish and sea turtles, and further on to visit Sea Life Park, the Sea World of Hawaii. Bring lots of cash, or just walk in (sometimes they don't notice) and tape a paper strip around your wrist as a white park wristband (worked for me).
On Saturdays and Sundays the Aloha stadium turns into a flea market and it's great for touristy shopping, but get there early. If you can't make there, head to the International Marketplace in the middle of Waikiki for some of the same trinkets, and have Dole Whip ice cream while you're there, and perhaps catch a hula show.
In the evenings from about 6-9pm, there are many street performers lining the sidewalks of Waikiki, from comedians to singers, artists, magicians and moving statues. Brings some tip money or just stand a ways back and wait for someone else to tip.
Chinatown is another place to buy unique, inexpensive gifts to take home, and also a great place to buy vegetables, or just take in some local scenery. If you're not used to Chinese markets, this will be an adventure. There are also many good places to eat there, both for authentic Chinese vendors and a variety of restaurants. Epic is my favorit Downtown restaurant. View their menu at http://www.restaurantepichawaii.com/. Try the butterfish and the sea weed salad. It is to die for!
A short trip up the Pali highway, stopping at the Pali Scenic Lookout, is the beautiful Kailua Beach, which is far less touristy, and much more peaceful than Waikiki. From here, you are not too far from the Ho'omaluhia Botanical Garden, a great place to see some of Hawaii's native flora and picnic by the water midst the backdrop of Oahu's green mountains.
If you have the money to spend on a theme park, you can travel back in time at the Polynesian Cultural Center to get a taste of the various tribal cultures of the south Pacific, and stay for the native theatrical dance performance in the evening.
The Dole Plantation is another unique attraction, which offers a pineapple shaped garden maze and a guided train ride tour of the plantation.
There are so many things to do and see in Oahu, and no way to fit it all in. I could go on and on, but this should get you started and give you a few ideas. Perhaps I will add more at another time. Aloha!