Since this will be your first trip to Aulani and Hawaii, I think you'll find lots of things to keep you busy. The resort itself has many amenities to occupy your time: the Rainbow Reef Snorkling, the Lazy River, the water slides, the pools, and the lagoon, not to mention the spa. You can book excursions at Aulani that will take to all over Oahu, including Pearl Harbor, Waikiki, and Diamond Head. Aulani also offers a for-pay shuttle that will take you to a couple of the major shopping areas. More details about the Aulani excursions can be found on the
official Aulani website under the Activities tab.
If you are going to be on Oahu for 11 nights, I strongly recommend you look into renting a car. Renting a car might save you money in the long run. Transportation costs between Aulani and Honolulu airport via taxi or shuttle will run $50-$75 each way, so a rental car could save you a bit. Also, with a rental car you can drive to the nearby SuperTarget and other stores to get groceries, drinks, snacks, etc, or drive to restaurants that are cheaper than the resort restaurants. Plus, with a car you explore Oahu at your own pace.
There is no charge to visit Pearl Harbor (which has the WWII Valor in the Pacific Museum and other exhibits. However, there is a small fee charged by the National Park Service if you want to tour the Arizona Memorial. You can book tickets ahead of time via the
NPS website. There is also a fee if you want to visit the USS Missouri. If you book a Pearl Harbor excursion through Aulani or another tour company, then these additional fees are included in the excursion.
There are many, many things to see and do on Oahu. There are so many things to see and do that I think your 11 days will go by very quickly. Eventually, you'll want to visit the other islands, but I would save them for future trips. Like Oahu, there are so many things to see and do on the other islands that its best to plan at least 3-4 days on each island.
Good luck, and e komo mai!