Why is it you don't want to rent a car? Renting a car would be the most economical option for touring. If you're staying on DVC points, parking is also free.
If its because you're apprehensive about driving in an unknown place, don't let that worry you. DH and I had never rented a car prior to our Hawaii trip, and we hated the idea of driving on vacation. It stressed us out and all the talk about the horrid traffic in Hawaii made us concerned. But after it was all said and done, we were SO glad we had rented a car and wouldn't hesitate to do it again. We loved having the flexibility to come and go wherever we wanted whenever we wanted. We were able to stop at Target to pick up groceries and toiletries we needed. We had access to tons more dining options outside the isolated area of Ko Olina. The highway driving wasn't as bad as I anticipated, and much of the driving is on one lane (each direction) roads with little traffic to contend with. We did only drive during the Aulani portion of our stay which included stops in Ko Olina, North Shore, and the windward side. We didn't drive near Pearl Harbor or Waikiki, and that would be more congested. We spent 3 nights in Waikiki prior to Aulani and opted not to rent a car then to save on parking costs and hassles. During that time we did organized bus tours to both Pearl Harbor and Diamond Head which worked out really well for us.
As PPs mentioned, without a car, your main option is taking the tours that Aulani offers. That puts you a set schedule, cost is $$$, and I don't think all of them offered even include transportation. Uber is also another option, but I don't think its all that reliable as availability from Ko Olina can be few and far between.