I grew up going to Hawaii for the summer since my uncle lived there so I know the place quite well. I've gone back many times as an adult with my family so here's my two cents.
As for public transport, I wouldn't count on that. The only thing they have in most places is local bus service, "DaBus", and it's not really set up for tourists. Definitely rent a car so that you can easily get to the beautiful beaches and valleys on your own time.
When we go, we use either VRBO or airbnb to find places to stay. There are some small hotels and B&Bs scattered around. One of our favorites is on Kauai in Kilauea North Country Farms Their two cottages are next to their organic fields and you are welcome to pick and eat whatever fruit or veggies you want. They don't serve breakfast but they do bring a breakfast basket which, for us, was all we needed! This is the only place we repeatedly go back to because we usually try to find different places to stay for a new experience each time we go.
Each island has a unique flavor so you need to decide which Hawaii you want to see! Oahu is pretty built up with the bustle of Waikiki, north shore surfing, and the long winding roads around the windward side of the island. Kauai is very lush and green, much less inhabited than Oahu. Most tourists stay in Poipu so once you get out of there it's not as overrun with bermuda shorts and sunburned skin. Take a sunset catamaran cruise around the mountains on the north end of the island - one of the best things to do in Hawaii! Maui is also less inhabited but with a strong tourist infrastructure. Lahaina and Kaanapali are the touristy areas. Drive up to the dormant Haleakala Volcano to watch the sunrise, drive the crazy winding road to Hana and count the waterfalls, or go out on a boat to see the Humpback whales. The big island of Hawaii is really big. The low parts of the island are where the tourists stay, the cowboys raise cattle and ride horses in the upcountry, and you can go see the active volcanoes in Volcanoes National Park. Lots of driving because of the size of the island but a very different place from the other main islands.
All of the islands have stunning beaches. My favorites are Ke'e Beach on Kauai (beautiful views of the mountains, sea turtles in the water, and roosters on the beach) and Lanikai Beach on Oahu (sugar white sand with multi-million dollar houses inhabited by movie stars). The further south you go in the island chain, the warmer the water, and the better the snorkeling.
No matter which island (s) you pick you'll have a great time!