If you really want to hit the most isolated bit of beach or waterfall, you might use 4wd (or rather the extra ground clearance), but most people will find far more than enough to occupy a week without getting off the paved or graded roads.
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The public (Hele-on) bus service is an alternative if you have plenty of time and want to avoid renting a car. The cheapest way to get around the Big Island after walking yourself is the public bus (Hele-on-Bus). This bus will get you to most places on the island, but transit times are long and connections infrequent.
However, driving in Hawaii does come with its own set of challenges that you should be prepared for, including poor road conditions. While the more populated islands like O'ahu or the Big Island have decent roads, the more rural isles like Kaua'i have roads that are often unpaved and difficult to drive on.
You can drive as far as the old Jagger Museum and get out to look over the crater to but there is so much more to the Kilauea Volcano. Expect to spend at least most of the day exploring the area and driving the Chain of Craters Road.
To discover Big Island as a whole, we recommend staying ideally 8 to 10 days so that you have time to explore it at your own pace, as the island is quite large.
Pros of Renting a Jeep in HawaiiFeel the wind in your hair as your cruise alongside the beaches. If you do a little off-roading, your vehicle can handle it. You won't have to use air-conditioning as the wind will handle that for you.