I haven't been to Maui. And you should know that I'm totally not a beach person - the only contact with water during my time on Oahu was while I was in the shower. And, I didn't have a car, so I was totally dependent on TheBus (Oahu's public transit system). I still had a great time. Here's what I did (that I remember off the top of my head):
Pearl Harbor
The Bishop Museum
Iolani Palace
Foster Botanic Garden
Hawaii State Art Museum
Doris Duke's Shangri-La
Honolulu Museum of Art (included with Shangri-La)
Waikiki Aquarium
Climb up Diamond Head
The "Circle Road" (by bus)
Various neighborhoods, including Waikiki, Chinatown, Ala Moana.
Of all of these, the highlight was Shangri-La. You must reserve in advance, as there are only a few tours a day and numbers are limited. Doris Duke not only had unlimited money, but great taste, and her home is amazing.
I also really loved the Hawaii State Art Museum. It was across the street from the Iolani Palace, and I only went because it was close and admission was free, but I was blown away by the number of high-quality works.
The Circle Road would be much better by car, so you can stop along the way. But even by bus, I could see the great variety of the island. From urban to suburban to rural, taking this ride really disproved the idea that there's "nothing of the real Hawaii" on Oahu. Certainly, seeing "Ching's General Store" showed me just how un-urban and un-touristy large sections are, as did seeing the places that looked just like New Jersey (strip malls included).
My most memorable meal for a good reason was at Okonomiyaki Chibo, where I was the only non-Asian customer. Lunch is much cheaper than dinner. I see they've moved since I was there in 2011. If you're curious about a kind of Japanese food rarely available in the mainland US, this is a fun place to try it.
My most memorable meal for a bad reason was at Rainbow Drive-In. This is famous as a place locals go to get local specialties. Indeed, I seemed to be the only non-local there. But it turns out the particular specialties served here are high fat, high carb, low taste junk. If you want junk food that's different from mainland varieties but just as bad for you, come here; otherwise, stay away.
A friend of mine did two things in Oahu that I didn't get to, but would like to for next time. One was a guided nature walk, and the other was a tour of Hawaii's Plantation Village http://www.hawaiiplantationvillage.org/. This was the highlight for him, as you get a guided tour from a former plantation worker and learn what their life was like. This is not to be confused with the Dole Plantation, which is quite different. Hawaii's Plantation Village isn't too far from Pearl Harbor, so you may want to try to see them on the same day (getting to these places from Waikiki by bus takes twice as long as by car, so it's quite a schlep to make that trip twice).
One issue I found was somewhat limited hours for these attractions. It was hard to fit them all in on my available days; do check hours carefully for any of your personal "must sees," so you don't miss them. For instance, Hawaii's Plantation Village guided tours are only at 10, 11, 12, 1, and 2.
TheBus website used to have a great guide on getting to various tourist attractions from Waikiki; they still have that text, but not the link. Keep looking - it was very helpful, as it's "turn by turn" directions (e.g., which corner to stand on and where to get connecting buses when needed, and how often they run).