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Cycling around the big island is great! I just did a 10 day trip around the entire island. Though I definitely agree that the Kona side is less enjoyable due to the resorts and the higher percentage of tourists, I wouldn't say that the roads are bad. They're more narrow in certain places compared to the other side of the island but I felt like those roads were safe and comfortable compared to a lot of what I experienced on the northern tier and pacific coast routes. I will say that the feeling of climbing out of the dry, lava rock Kona side and then suddenly dropping down into the much more lush and green Punalu'u area was incredible. You get a similar feeling as you leave Volcano national and drop into Hilo or Puna. Landscape and scenery changes quickly and dramatically
I flew into Kona and rented a mountain bike and a BOB trailer from Kona Bike Works. It worked out fine (especially on the dirt roads and trails that I chose) but I definitely missed my touring bike as I'm not used to flat bars or plastic-based saddles.
At first I just used the free map I was given on the plane but eventually purchased a map from the welcome center at Volcano National Park. (It's this map: http://www.amazon.com/Map-Hawaii-Island-Reference-Islands/dp/0824826418).
I camped out every night at county parks except for 3 nights. It's $5 for a permit ($6 online) and I found that was the cheapest option. http://www.hawaii-county.com/parks/parks.htm (I think Spencer might be the only place where they check permits but I didn't mind supporting the county parks.)
I also stayed at the Hilo Bay Hostel (really nice), Hotel Honoka'a Club (not as nice but close to Waipi'o and Honoka'a is a great town), and I camped one night in Waimanu valley which was a bit of a hike, though my favorite part of the trip.
Definitely go down and see the farms/beaches/surfing in the Pahoa/Puna area and also the Kohala coast in the north. I think these are both areas that cyclists frequently skip since they're essentially side trips that can easily be avoided if your goal is just to go around the island.
Keep in mind that there's lots of climbing on that island! -- but it's worth it. Hawai'i is such an intense place and it's sad to think about all the people that travel there and leave without any sense of what life is like there.