With the exception of Oahu there's really no public transportation on any of the other islands so having a car is really the only way to get around. Many people ask me if it's really necessary to to have a car on a Oahu. In my opinion, the answer to that question is yes. It's always helpful to have a car even on Oahu because most of the time you're going to be staying in Waikiki which is a very busy metropolis with lots of action and lots of people. If you want to experience the real beauty of Oahu then it really is a good idea to rent a car and drive to the many beautiful parts of the island that you never see in Waikiki. For example, you can drive to Lani Kai Beach which is one of the most beautiful beaches in all of Hawaii or drive to the east side of Honolulu to Sandy beach or Beautiful the beautiful white sand of Waimanalu.
If you don't want to rent a car on a Oahu for financial reasons or because maybe you're underage (you need to be at least 21 years of age to rent a car in Hawaii) it is possible to have a nice experience but if you can rent a car at least some of the time while you're on Oahu, it's really a good idea. As far as the other islands go, if you cannot rent a car on Maui then you need to stay in the Lahaina area, preferably someplace near the front Street area becauce there is a lot to do and there are small shuttles to and from the hotels if you just don't have a car. There is also transportation that can be arranged from the airport to the hotels but it is a bit costly. The same goes for Kauai, however there are no shuttles anywhere however you can get by staying in the Kapaa area without a car. Likewise, on the Big Island you can survive somewhat without a car in Kona.
Having a car on any of the islands is certainly something you should consider if not making it part of your vacation for sure.