Port Moresby is located at the edge of the Papuan Barrier Reef, which is full of wonderful diving sites. There is a dive center on Loloata Island where you can rent diving gear but you have to be certified diver.
There are a number of nightclubs in the city. Some of the better-known ones are the Royal Papua Yacht Club, the Cosmopolitan Club, and the Gold Club. Please go with a partner for safety. Gambling is legal in Papua New Guinea (legalized in 2007), so there are a number of casinos and gambling houses in the city. Even then it is better to stick to the larger, reputable casinos.
Other interesting places nearby are Sogeri Plateau (which is the end of the infamous Kokoda Trail) and Yule Island. Yule Island has located two hours away from the city. The place has a distinct Catholic influence and is good for a quick getaway. They also serve great seafood on Yule island.