Honolulu is the major city of Oahu. If you are in Waikiki then you are actually inside of Honolulu. Honolulu ruffly starts at The International airport all the way to Diamond Head. The next Major city next to Honolulu would be to the east; Hawaii Kai
Hospital
There are a couple clinics in Waikiki, however, the nearest full service hospital would be:
Queens Medical Center 1301 Punchbowl St · (808) 691-1000
Airport
Daniel K. Inouye International Airport:
300 Rodgers Blvd, Honolulu, HI 96819
Phone: (808) 836-6411