The best way to get around the park is to drive yourself. You could possibly hike part of the park but if you are going down the chain of craters road you for sure need wheels. That said we did see several on bikes on that road but they were in training it appeared.
It's cheap and easy to rent an economy car and drive from Hilo to Volcanoes National Park. The drive is very pleasant and the highway in excellent condition. In our case, we left in the morning and drove along the coast highway, stopping to walk at lava beach for awhile, before driving inland and up the mountain. At the park entrance, the Volcano Village museum and bookstore are worth another hour, but keep moving. Take the caldera rim road, making stops wherever the view seems best, and consider walking through the volcano tunnels. We found the chain of craters road perhaps the most scenic, but the ultimate goal for most coming to the park will be a live viewing of molten lava. Park the rental car where you need. We returned to the car after dark, and then drove quickly up the chain of craters road to the National Park's hotel. By the time we arrived, the restaurant was already closed, but we managed to get some food to go...
Becuase I had limited time, I organised a day tour of Big Island from O'ahu. I guess you could get flights and a hire car and do it yourself, but honestly I just felt like going along for the ride. The tour I booked was a one day Volcano tour. The company organised pick up from my hotel, transport to the airport and then a full day tour on Big Island. It was a great day and worth the money in my opinion. (US$260). When else would I get to see live volcanoes? The guys I was travelling with decided to go surfing instead....again!
I caught a flight from O'ahu to Big Island. It only took about 1 hour and was quite pleasant. It was a great way to see all the islands from the air. This photo is of Maui. Unfortunately for us, we flew into Hilo on Big Island and didn't go past the Volcanoes. I would have loved to see them from the air, but I guess we saw plenty of extinct volcanoes on all the other islands as we flew over.
By plane from Oahu on the Polynesian Adventure Tours: http://www.polyad.com/popuptours1/oahuh2.html