The hike up to Diamondhead was one of the highlights of our honeymoon. It's beautiful, both the views of the Pacific and the city from the top, and the park on the way up. It's modestly strenuous as a hike, probably less so if you're an active hiker.

The aquarium near Waikiki Beach was also quite fun.

Foodwise there exist lots of options which cater to tourists in the area we were. Those were mostly pretty good, but largely in the fashion that you expect for restaurants catering to tourists. I strongly recommend you get a recommendation or two from somebody local. (We did for a meal or two, but unfortunately I don't remember either of the places well enough to Google or name them.) The single best restaurant we went to was Roy's, which does Asian-fusion among other things. It was good enough to build a trip around, and Ruriko and I will probably do that at some point. There is also a Nobu, which is pricey, but offers really credible sushi. (I've had good sushi in the US, but Nobu is the only sushi which would tempt me to cheat on my local sushi joint in Japan.)