The only way to go from one island to another is by airplane. Check with all the different airlines - Hawaiian, Mokulele, Go!, Island Air, Pacific Wings, etc on what their bike prices and rules are. Some of the puddle-jumpers may not take bikes.
On Hawai'i our roads are narrow with blind curves, bad weather and often without shoulders. There are bike tour places you can call about possible routes and maybe even rentals. There are very few places that will rent bikes. Hilo Bike Hub in Hilo may be able to make route suggestions for East Hawai'i. (no rentals) However, I can't imagine why you'd actually want to ride on the highways here. Some of the more interesting roads like Kohala Mountain Road and sections of Old Mamalahoa Highway are also some of the more dangerous. The upper cane haul road between Pahala and Na'alehu might be interesting, but be aware of a long stretch of cobble and unpaved on the Na'alehu side. The best bike option on Hawai'i would be taking a mountain bike out on some of the mountain roads, in the timber plantations, and on old cane field roads, but you'd need a car to tote the bike around from place to place with. (These places won't be very well mapped, though.) Watch out for dogs, many of which are hunting dogs trained to chase and grapple/kill pigs. Pit bulls are popular. "Local" people can be very territorial at times. You will see a lot of "no trespassing", "beware of dog", "private road", "keep out" and "kapu" signs.
Finally, Do Not Bike On Saddle Road!
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-=- '05 Jamis Nova -=- '04 Fuji Absolute -=- '94 Trek 820 -=- '77 Schwinn Scrambler 36/36 -=-
Friends don't let friends use brifters.