Winter is not the time to be trying to enter the Honolulu rental market unless you absolutely have to. This puts you in competition with thousands of people that move here from the rest of the country in order to enjoy the year-round beautiful Hawaiian weather. Instead, start looking in April and May, when many of the winter residents are starting to vacate.
When renting in Honolulu, remember that the demand for apartments in paradise is very high indeed. You need to think about what you can do that will set you apart from other people trying to rent a given apartment. If you have good references, provide an extra one over what you're asked for. If your financial situation is very good, provide a letter confirming your employment. These things can all help edge you over the others who want the same apartment.
Driving in Honolulu is not for the faint of heart. If you do decide to drive, you should invest in a high quality GPS. Otherwise, you run the risk of getting easily confused by the convoluted street layout, and the local predilection for giving directions based on landmarks instead of compass directions. The Bus service known as "TheBus" offers good, affordable service not just in the city, but across other parts of the island as well.
Hawaiian culture and sport are both heavily focused on surfing, and if you've never tried it, this is your best chance. Waikiki is the most famous beach, but there are several others that might be less crowded with tourists. There are several memorials at Pearl Harbor that commemorate that terrible day in the county's history. You can also tour the Lorain Palace, which was the home of the last two rulers of Hawaii. In the district that lines Waikiki beach, and to the immediate south, you will find the largest selection of bars and restaurants in the city.