Some places I saw and enjoyed on Oahu, in addition to Pearl Harbor already mentioned:
Doris Duke's Shangri-La: this is her former home, now seen by small group tours (only a few per day). Unfortunately, I see from their website that it is closed for much of November (reopens November 23). If it's open for your dates, book now to ensure a spot. She had both unlimited money and great taste, and seeing this was a real highlight for me. It also includes admission to the Honolulu Museum of Art, which was good (probably not worth a special trip on its own, but nice nonetheless). http://www.shangrilahawaii.org/visit/guided-tour2/
A few blocks from the Honolulu Museum of Art was a Goodwill. If you want to buy Hawaiian shirts or polo shirts, this place has a larger selection than any regular store, and they're cheap. The address is 1075 S Beretania St. I only went in because I had some time to kill before my Shangri La tour, but I was dazzled by the shirts - racks and racks of them! Of course, there are lots of places to buy new Hawaiian shirts, at higher prices.
The Iolani Palace is the only royal palace on US soil. And cater-corner from it is the Hawaii State Art Museum. I only want to the museum because it was close by the Palace and free, but I LOVED it, and would definitely go again even if it weren't free!
The Bishop Museum had some interesting exhibits on Hawaiian culture.
There are several botanic gardens; I went to the Foster Botanic Garden and enjoyed it very much.
Inside the Moana Surfrider (the oldest hotel in Waikiki) on the second floor is a free exhibit with some artifacts about early Waikiki tourism. In addition, you can go out on the second floor balcony for free and sit overlooking the "main drag." I also just liked soaking up the atmosphere at the Surfrider; it has much more of this than the more famous "pink palace" (the Royal Hawaiian).
My most memorable meal was at the Okonomiyaki Chibo. Okonomiyaki is a "Japanese pancake" (it's actually quite different from a US pancake), and is hard to obtain outside of Japan. I was actually the only non-Japanese customer (the staff is bilingual), so it was a real cross-cultural experience. Note that their prices are MUCH lower for lunch than for dinner - for the same food. When I was there they were in the Royal Hawaiian Center, but I see from their website that they've moved (still in Waikiki): https://chibohawaii.com/
I didn't get to do this, but a friend highly recommends the docent led tour of Hawaii's Plantation Village. Unlike the Dole Plantation (which looks like a theme park), at this one you hear from an actual former plantation worker about life on the pineapple plantation, which was operating until the early 70's. Note the limited times for tours, and also note that it's close to Pearl Harbor but far from Waikiki, so you may want to plan carefully if you're taking The Bus instead of driving. http://www.hawaiiplantationvillage.org/
One issue I found was limited hours for these attractions. Many are only running from about 10 to about 4, and including commuting times from Waikiki, it's hard to see multiple ones in one day if you don't have a car. However, it's definitely worth seeing at least a few of these, as they get you outside of the Waikiki bubble (or as I call it, "Times Square with a beach").