Oahu has a good bus system, but the other islands do not. On these other islands you will need a moped or car. When you arrive you can usually rent a moped for cheap until you buy your own transportation. Some just rent the moped for their whole stay.
I would recommend getting a moped. It's cheap and it's pretty nice to cruise on since it's warm. Unless you have a/c or a sunroof I found it more comfortable to ride on a moped mid day unless I was going very far. Since I didn't get stuck in traffic it took about the same time for me to go to school on the moped or in my car. And driving around in Waikiki was definitely faster on the moped and no need to look or pay for parking.
Mopeds do not require insurance. If however you buy one that the state knows is faster than 30 mph you will need to register it as a motorcycle, get insurance and a motorcycle license. Mine was faster, but at that time they did not care.
I bought a Honda metropolitan with 800 miles on it (about 1 300 km) for $1650. When I left i sold it for about $1000 and I was happy with it during my 2.5 years in Hawaii.
When I worked as an instructor I also bought a car. If you buy a car you need to check that the head gasket is ok. It is a quite common issue among old used cars especially on the islands with steep slopes like the big island. Three easy checks are: check that there is no oil in the coolant, check that there is no water in the oil and drive the car uphill a for a few km. If it makes it up a steep and long slope with temp under control, it's good.
