First off, especially if you're going to Oahu, getting places early is often important. When we visit, we try to keep from getting fully adjusted to the time difference as long as we can, so we have an easier time getting where we're going before the crowds and Honolulu traffic. My experience is limited to Oahu, so this might not be quite as important on the other islands. If you end up there, several great options have already been mentioned, but here are some of my favorites.
Pearl Harbor - It's a must in my book. (Get there early)
Hanauma Bay - Whether you snorkel or not, it's stunningly beautiful (again, get there early). Finish up at Hanauma Bay around noon, and you can spend the rest of the day exploring the southeast corner and windward side of the island. Every inch of it is amazing, and even if you do nothing more than take in the views, it's worth it. The hike to Makapu'u Lighthouse is worth it, in my opinion.
North Shore - You can easily spend a whole day here. Visit Haleiwa and HMS for sure. Head northeast from there along the North Shore and stop off at Laniakea Beach to hopefully catch a glimpse of a Honu (Hawaiian Sea Turtle) or two (around lunch time is best). From there, keep heading northeast and be sure to stop and take in the pipeline. Regardless of the time of year, the waves are awe inspiring, at least to landlocked folks like me. I can sit and watch/listen to them crashing for hours, but don't even think about getting in the water. If you have little ones, don't even let the water touch them. Even water up to their knees can suck them out in the blink of an eye. That is no exaggeration. Grab lunch farther along the North Shore at Seven Brothers in either Kahuku or Laie (I've only been to the one is Laie). They have, without a doubt, the most amazing hamburgers I've ever tasted. Another option is to head west from Haleiwa, past Dillingham Air Field, and watch some amazing kitesurfing at Mokule'ia Beach. There are plenty of other spots on the North Shore as well, like Waimea Bay and Waimea Valley and Pu'u O Mahuka Heiau State Historic Site that I'd recommend as well, but it may require two days to hit them all. (If you can't tell, I really like the North Shore.)
Iolani Palace - It's a great place to start learning some of the history of the Hawaiian people and what they've gone through.
Pick a trail to a waterfall (a legal one) and take it. There are a ton of options.
Find the perfect point to watch a sunrise from, somewhere relatively secluded if possible, get there an hour or so beforehand, and experience peace like you've only dreamed of. Do the same for a sunset. There are apps you can download to find the exact time and direction of sunrise and sunset from any spot on earth. Finding the perfect spot is half the fun.
Last but not least, take a trip to the Waianae Community Center to pay homage to the memory and legacy of Bruddah IZ. There's a bronze bust of him there.
I'm just an occasional visitor to the islands, so my knowledge is limited. However, the memories I've made at these spots will stick with me forever.
Aloha,
Matt