There is a decent bus system on Oahu. It'll take a lot longer to get places than it would by car, but it's an alternative.

http://www.thebus.org/

e.g., 42 minutes from the Hyatt Regency Waikiki to Diamond Head.

Oops, should have added, "as opposed to 6 minutes by car".

My first several trips to Oahu were car-less ones. I relied on the all-day pass on the trolleys. That worked just fine for me. And yes, the bus goes everywhere! We jokingly said we were going to take the bus around to the north shore and just get off and not get back on (as in, just STAY on Oahu)... hahaha. Traveling by bus is time consuming, but it can be done.

Now, my annual trips are WITH a rental car. I like having the car to make the drive around Oahu for fresh shrimp at one of the shrimp trucks on the north shore. I like driving to the malls and having a car to throw packages in!

But that said, parking a car can get expensive in Waikiki. I just bite the bullet and do it (usually for a week at a time).

No you don't need a car for that you describe. "The Bus" (the normal public transportation system) is what you would use, or taxis.

The trolleys are more just a fun ride-around for tourists and shopping, etc. They don't go to DiamondHead or PearlHarbor (I don't think?).

People who want to get out of Waikiki and go around Oahu often like to rent cars, either their entire stay or a day or two.

Within Honolulu you can walk or use The Bus. Even a taxi won't break the bank because distances are short. I would always recommend a car for at least one day to tour Oahu's beautiful east and north shores. For a short visit, I'd recommend picking up a car at the airport. You don't want to waste time on busses, and you have to get to and from the airport anyway. (Yes, you'll pay to park at your hotel, but it's probably a teeny piece of the cost of your trip.)