I lived on Oahu for six years, and I can tell you how to get to places that are totally away from the tourist areas - all will require hiking though. In fact, the best suggestion is to find a book on hiking Oahu - I'm sure there are several that you can find on Amazon.
Here are several tips:
Waianae Mountains - west side:
1. hike up to Kole-Kole pass - this is where the Japanese airplanes came in on the attack.
2. There is a terrific hike that goes to the top of the highest part of the Wainae range; can't remember the name of it now.
Koolau Mountains - east side:
If you look towards the Koolau mts from Honolulu, you will see several "fingers" with houses going up the mountains. Almost every one of these has a hike that starts at where the houses stop, and goes all the way to the Koolau ridge, looking out over the North Shore. The two best, and shortest, are the ones that 1) take off from the top of St. Louis Heights (near the university) and 2) from the top of Wilhemina Rise. These are two of my favorite hikes, and there are fabulous landscape photography opportunities.
Hope this helps.