I am assuming you already know about surfing lessons, Pearl Harbor, Iolani Palace (the only royal palace in the USA), so here are other items you may not know about:
OBON: If you want something totally different, you are going during "Obon" (pronounced "oh-bone"), the Japanese festival where deceased ancestors come home to visit their families. Japanese-Americans are a large majority of the population in Hawaii, this is an important festival at many buddhist churches throughout Oahu. Ask your concierge or check on the internet for dates by searching for "Obon 2009 Oahu". The best one (I think) is in Manoa Valley, in the back of Honolulu on July 31/Aug 1st.
http://www.gohawaii.com/event?id=1597
JUNGLE HIKING: For hiking in the rain forest, don't wear shoes (will be unusable afterwards) but make sure you bring good quality flip-flops or Tevas (very muddy, and I see a lot of broken cheap flip-flops on the trail..how do people walk barefoot through the jungle??). And wear repellent. Manoa Falls is right in back of Honolulu, is the highest falls on the island, a small tropical pool at its base, you hike through bamboo forests and giant ferns to get there. Mauna Wili Falls on the windward side is also great, not as lush/slightly drier and bordering on "moderate" hiking. A lot of people hike Diamond Head Crater. I don't particularly think it is a good hike: the Diamond Head hike is in the center of the crater up the rim wall from the "inside", it is hot and dry since the rim of the crater blocks the wind. The view is spectacular at the end though.
http://www.hawaiiweb.com/html/hiking/man…
CHINATOWN: Make sure to walk through the downtown chinatown center. Go along Mauna Kea street to watch the ladies string flower leis where you can buy a fresh orchid lei for about $7. I like Cindy's Lei Shoppe (well-known) but you can select one you like from any of them...note dendrobium orchids last the longest, some flowers (ginger, plumeria) smell good but last only a day or two. then go into the Mauna Kea Marketplace entrance on King Street (it is an inside food vendor mall). If you have never been in a tropical food market in Indonesia or Philippines, this will be interesting to you. An amazing food court.
http://www.cindysleishoppe.com/
BUDDHIST TEMPLE: Have you googled the buddhist Byodo-in Temple on the windward side? Especially if you have not been to Japan, really worth seeing, and usually not too busy mid-week. Ring the bell for good luck. Don't be put off by the entrance road (you have to drive through a cemetary!).
http://gohawaii.about.com/od/oahusights/…
NUUANI PALI POINT: This is considered to be one of the most spectacular view in all the Hawaiian Islands. Take the Pali Highway in back of downtown Honolulu towards the windward side, exit at the lookout. The tradewinds funnel through the mountains here, can be amazingly windy, sometimes dangerously so, just wear sunglasses (part of the fun).
WINDWARD DRIVE: Rather than any destination, the DRIVE itself is what is important here, starting in Kaneohe and ending up on the North Shore. Honolulu is on the drier part of the island, so it is only when you go to the windward side (the side of the island where the wind comes off the ocean and drops the rain) that you will say AH I AM IN A SOUTH SEAS PARADISE! You will pass the Polynesian Cultural Center on the way. This has pavilions dedicated to each major polynesian island in the South Pacific (Fiji, Tahiti, Samoa, Tonga, etc.) They also have a luau (the Center is owned by the Morman Church, so no mai-tai's/booze if that is important to you).
While you are at Pearl Harbor, the Arizona Memorial is a tear-jerker of course, but pay the extra $$ and tour the fully-renovated WWII submarine in the same area. I am not a military history junkie, but it will blow your mind how they stayed underwater in such tight quarters for so long.
SNORKELING: Of course, you may want to snorkel to see the coral lagoons, tropical fish and possibly see a sea turtle. The most famous is Hanauma Bay (east of Honolulu, past Hawaii Kai, going counter-clockwise along the coast). This is internationally-acclaimed. Rent snorkel equipment from snorkel bob's, or rent from the concession on the beach here. Go early, parking is limited and fills up quickly.
NORTH SHORE: Aside from Waikiki, the most popular beach is Waimea Beach on North Shore. Go up and view the famous beaches (Sunset, Waimea, etc.) then get a Hawaiian Shave Ice in the surfing village of Haleiwa (pronounced "hah-lay-ee-vuh") at Aoki's or Matsumoto's.
SHOPPING: Waikiki has interesting shopping, and certainly look at the huge banyan tree at the International Marketplace...but for "real" shopping at a reasonable price, try the Ala Moana Shopping Center. This is a mega shopping center that includes both souvenir and major stores like Macy's, Neiman Marcus, Nordstrom, Sear's...you get the idea.