Aloha! I live on Kauai and for obvious reasons would recommend visiting Kauai even if just for part of your trip. I lived on Oahu (in Waikiki) for 3 years and would recommend only visiting there for a couple of days just to see the major attractions such as the Arizona Memorial. Oahu is very crowded and the traffic is terrible.
Contrary to what others have noted, there is public transportation on Kauai although it isn't the most reliable...we do have a bus. Also, there's always Uber! Yes, Kauai actually has Uber. I've used it on one occasion.
Also if staying on the south side of Kauai (Poipu Beach area) there is a complimentary shuttle bus that will take you from your condo/hotel to a restaurant if you don't want to have to walk too far. It's nice if you have a drink or two with dinner and don't want to drive home....or if you don't happen to have a car that day. You can also get your rental car in Poipu too, instead of at the airport in Lihue so there's always that option to consider.
Just remember this, when you arrive in Hawaii....slow down....take it all in but don't be in too big of a rush. :)

Mermaid, while there is a bus on Kauai, it really isn't practical public transportation for visitors. I know someone who tried to use that bus for transport when visiting Kauai - after one day these very patient people opted to rent a car. We spend 4-5 weeks every year in Poipu, and I have never seen or heard of the shuttle you mention. IMO, if you want to go to Kauai, you need to count on needing to rent a car.

While looking at the overlook near the St. Regis Hotel in Princeville, I recently chatted with a woman who was staying at the Kauai Beach Resort, near the airport. She had taken a bus to the Princeville shopping center and walked a few miles to the St. Regis. Her plan was to walk back to the shopping center and take the bus to Hanalei. From there she would walk at the beach, do some shopping, eat and then take the bus back to her hotel. So, it is possible to get around Kauai by bus, but it sounded like she would be spending hours waiting bus stops, and doing lots of walking. There is also the possibility of taking day tours to places like Waimea Canyon.

But for me, the inconvenience of not having a car on Kauai would not be worth the savings.