Hilo is the Big Island's big city, green, luscious and properly tropical, unlike other parts of the island overrun by volcanic activity in the last century. With a handful of manageable cultural activities and easy access to the rest of the island, Hilo is a good place to start your adventure.
Hawaiian history buffs will want to stop and tour the Lyman Mission House and Museum and the Pacific Tsunami Museum—Hilo has had two big tsunamis in the last century—for artifacts and relevant artistic works. Get your daily dose of geology at the Earth Heritage Gallery, stopping to see the replica of a Hawaiian grass house in the same building.
Anyone with Jungle Book fantasies can at least enjoy tigers in the rainforest, even if they are behind bars, at the Panaewa Zoo. This rainforest didn't get here for lack of showers from above, so come prepared to experience some of Hilo's 130 inches of yearly rain. Visit the Hawaii Tropical Botanical Gardens just out of town for more cultivated greenery without the big animals, or the Nani Mau Gardens.
Because of the regular and recent volcanic action on Hawaii, there are disproportionately fewer white sand beaches here compared to other islands in the chain. Get a bit of dark sand between your toes at one of the beaches east of Hilo along the road to Leleiwi Point. Try snorkeling at Richardson's Beach Park while surfers can enjoy the winter swells at nearby James Kealoha Beach County Park. Anyone interested in doing some deeper dives into the sea can rent tanks locally or get scuba certified with a few days of lessons.
For rainbow-rimmed waterfalls and some rushing river, head in the direction of Wailuku River State Park along Route 200, stopping along the way to capture the cataracts on film.