I used to have family in Hawaii, so I went a lot growing up, and actually, went fairly often until I had kids. We just took the kids for their first trip last month, and DD3 was terrible on the flights, so we're not going back for at least a couple of years.
We loved Aulani.
We did all our Oahu sightseeing in a couple of days before we headed to Aulani, then spent five days there without leaving the resort (except to take DS to the ER and spend a day with him in the hospital, but that is another story). I would definitely plan on a minimum of 5 days relaxing at Aulani, and I would spend more time there if it was my only base for sightseeing on Oahu. You definitely want to go to the PCC for a day, and you really should drive over to north shore and grab a meal of fresh shrimp off the shrimp trucks and shave ice. (There is shave ice at Aulani, too, and it was good, but I'm not convinced it is the best on the island.) Since it is a first, and probably only, trip, you probably also want to go to Pearl Harbor, go by the Aloha stadium swap meet for some shopping, and visit (and possibly hike, depending on your group's stamina and interest) Diamond Head, and visit (and possibly snorkel) Hanuama Bay. I might also try to do a whale or dolphin watching boat ride or a submarine tour. The Atlantis submarine ride we did last month (just off Waikiki, so not as spectacular as some of those that can be done in other parts of the islands) turned out to be one of the highlights of the trip for both my DH and DS. We saw a huge number of variety of fish, 4 sea turtles, and 2 sharks.
We did not island hop this trip - since we were traveling with a 3 year old, we decided to hop to California on our way to and from Hawaii instead, to break up the flights. We have island hopped on almost all our previous trips. If this is a once in a lifetime trip, I might actually do a lot of hopping.
A couple of things you might consider, if your interest and budget allow -
1) One day to Kauai. Kauai is small, and you can actually hit many of the many scenic spots in a one day tour, and there are several groups you can book these through. It saves you the hassle of packing up and moving for just a night or two, but gives you a taste of the beauty and uniqueness of Kauai.
2) Several days on Maui with a snorkel tour to Lanai - my father and step-mother considered the day trip they did to Lanai the highlight of their trip. There is a ferry that you can take from Lahaina (in Maui), or you can book an excursion that includes the snorkeling.
3) A few days on the Big Island. Go to one of the black sand beaches, Volcanoes National Park, the visitor center at the Observatory, and the banyan trees, Rainbow and Akaka Falls, and the tropical botanic garden in the Hilo area. There is so much to do on the Big Island, and it is very different from the other islands.
No matter what you choose, you'll see a lot of beauty. You really can't go wrong.
There is a lot you can do and see in the amount of time you've allocated.