I lived on the Big Island for many years (miss Hawaii). I can't say too much about the logistics of Koko crater and Diamond Head, but as far as fun things to do on Oahu:
Rent a jeep and go to the North Shore - also just take a drive around the island, period. There's lots to see. (Be prepared for gnarly traffic, however!)
Visit the International Market off Kuhio Ave down in Waikiki - lots to see there
They have a zoo off Waikiki towards Diamond Head
Visit Pearl Harbor to view the memorial
Be sure to try out local restaurants - if you like seafood, definitely try the Ono. Hawaii has a very large Asian population, so you're also pretty much guaranteed to eat amazing Asian cuisine there, like Thai, Japanese (sushi!!), Korean, etc.
As far as transportation, I do believe they have some shuttles that run up and down Waikiki, so check for those - I think they're called Hele-on
Again, I lived in the Big Island so I didn't get over to Oahu much. When doing so, we typically took advantage of Honolulu's city atmosphere because that wasn't available in Kona. So a lot of the things we used to do involved city-type experiences, such as visit Ala Moana mall, hit the Cheesecake Factory ('cause again, they don't have one on Hawaii island), etc.
Hopefully someone else has some other fun suggestions for you. Have a great time!