Hawaii is perfect for that. There is STUNNING hiking on Kauai, Maui, and Hawaii (the big island). I would recommend Maui and Hawaii in particular because they both have national parks. However, the Na Pali coast on Kauai and Waimea canyon are breathtaking hike as well. Basically, you can't go wrong here. The volcano on Hawaii gives a totally surreal, unique landscape. And Haleakala crater is amazing too.
If you go to the big Island of Hawaii, spend a couple nights near Volcano National Park. Volcano Hideaways are lovely cottages, if you are lookng for a place to stay http://volcanovillage.net/. Volcano is a super quiet town, with not much to do other than see the National Park. It is not particularly near to nice swimming beaches or anything. So I wouldn't recommend spending your whole honeymoon there, but it's got a very romantic and secluded feel for a few nights.
Don't miss the snorkeling if you like to swim! I could spend hours watching the fish near the reefs on Hawaii. Ask locals for the best snorkeling spot. Or, rent kayaks and take a self-guided snorkel tour. My fiance did this on the big island (Hawaii) last year, I think near Captain Cook, and LOVED it. He did get a sea urchin stuck in his foot though, so watch your step!
Last piece of advice: rent a convertible. Totally worth the extra cash, if you have a little room in your budget. And Hawaii is perfect for it. Have fun! :)