We didn't have much time on Oahu so this post won't be very long or comprehensive.
Arizona Memorial at Pearl Harbor : The memorial was solemn and educational.
Iolani Palace : Also historically interesting and very grandiose. Nice garden areas outside, also.
Hawaii State Art Museum : I really enjoyed the exhibit that was there "Hawai‘i: Change & Continuity." It opened in Sept. 2015 and the website doesn't list an end date and features artwork by Hawaiian artists. The piece in the center is an art quilt by Karen Gally.
In the same building as the Hawaii State Art Museum is the HiSAM Sculpture Garden which features two interesting art pieces utilizing glass and colors. Above is a pool that has been painting with glass tiles on top of the paint. (That's me "swimming" in it).
This piece is above me in the photo. It's a pavilion with glass on top that has been painted. The colors on the concrete around me are the sunlight shining through the painted glass.
Kuhio Beach Torchlighting and Hula Show : Free shows on the beach on most Tuesdays, Thursdays, and Saturdays.
Beaches : There are several beaches on Oahu. In Waikiki we were able to walk a few miles along the beach going by the backs of the hotels.