This is going to be an ongoing list, but for now, I'll start with a few.
- The hike to the top of Diamond Head (parking costs about $5, but you can walk in for free).
- Pearl Harbor's USS Arizona Memorial (though the nearby submarine USS Bowfin and the USS Missouri cost money).
- Hanauma Bay, go snorkeling with an amazing array of tropical fish and the occasional sea turtle (entrance is $5 but free if you're kama'aina [local resident], and the snorkeling equipment can be rented for $5 or so if you don't have our own).
- The self-guided Oahu tour of Lost (well, I guess this will cost for gas).
- Whale watching (during the winter months) from the paved trail to the lighthouse near Makapuu, the easternmost point of Oahu (from which you can see Molokai and even Maui, and supposedly the Big Island).
- Any state beach, as well as Waikiki.
- Watching tourists along Kalakaua Avenue (both the shopping district and the part that runs along the beach).
- The nightly fireworks display from Waikiki (and the off-shore spectacular every New Year's Eve).
- The Halloween Parade in Waikiki, Honolulu's answer to Mardi Gras (wild costumes that are inappropriate for children).
- Kauai Coffee Company (free samples and a free tour). I was expecting a bit of a pricey tourist trap — cough! Dole Pineapple Plantation on Oahu cough! cough! — but this was the real deal and all free.
- Waimea Canyon.
- Sprouting Horn blowhole.
- Wailua Falls, the double waterfall shown in the opening of Fantasy Island.
- Any state beach.
- The hike along the Na Pali coast, starting at Haena Beach Park and going west.