You will need a car on the Big Island. Get a four-wheel-drive vehicle if you're at all interested in exploring. Some of the island's best sights (and most beautiful beaches) are at the end of rough or unpaved roads. Talk to the agency in advance if you want to pick up a car at one airport and drop it off at the other. Though they allow this, most charge an additional fee of up to $50. If you arrange it ahead of time, they can often be talked into waiving the fee.
Most agencies make you sign an agreement that you won't drive on the Saddle Road, the path to Mauna Kea and its observatories. Though smoothly paved, the Saddle Road is remote, winding, unlighted, and bereft of gas stations. Harper's, a local rental company, is the sole exception.