Aloha! The Big Island of Hawaiʻi gets its name from its sheer size. The best mode of Big Island Transportation is by far renting a car. (However, if you’re just coming for a day trip, there are tour companies that will pick you up and drop you back off at the airport.)
There is a bus system, Hele-On Bus, that will take you from one side of the island to the other. It runs once a day. See below for details.
The Mamalahoa Highway makes a complete circuit around the Big Island. You can drive it in one day, if you don’t plan on making very many stops. This isn’t recommended; you’ll want to stop for sightseeing. There is also Saddle Road that runs across the middle of the island. It is called Saddle Road because it courses over the “saddle” between the very large mountains of Mauna Kea and Mauna Loa. This is a high altitude road, and many rental car companies have policies against driving over it.