As far as getting around once on the islands, really the best way to go is to rent a car. If you are on Oahu and do not plan on leaving Honolulu then feel free to use taxis. They can be expensive, but no more than they would be in San Francisco or Manhattan.
There is also an adequate public transportation system on Oahu if you don’t mind riding the bus.
There is the Speedi Shuttle service that is on Maui and the Big Island and has a fixed price. Make sure to call in advance to find out exact availability.
Most all rental car agencies are available on all islands. Be advised when driving on the islands to slow down but not so much that you are holding up locals from getting to work. The speed limit on some islands is much slower than the average freeway so enjoy the scenery and if you see something you want to enjoy or are confused, please pull over.
Maui also has a small bus that helps you to get around the island. The bus is small and only goes to specific areas so it is not ideal for everyone. But it does help you to get from point A to point B in the smaller towns, if needed.
The Big Island is vast and sprawling and has least an hour in between towns. There is a bus system and can take you to other towns if you don’t mind the wait. I would not advise using taxis unless staying only in Kona or Hilo.
To really enjoy this island it is best to have a car in order to truly enjoy the diversity of its environment.
Kauai also has a bus system and once again only goes to specific areas. This islands lush scenery can not be enjoyed on a bus route, but if that is what is needed, then so be it. Be aware that you can only drive around half of this island and the other half is accessible by hiking or boat.
As noted before, the best way to see the islands is by renting a car unless you plan on staying in the city limits of Honolulu, then taxis would be appropriate. Some major hotels do have shuttle services so check with them as you make your reservations. When you drive into the smaller towns and scenic spots, you are able to get a taste of old Hawaii and you will be reminded of why it is you chose these islands as your travel getaway.