Port Moresby is the main point of entry into Papua New Guinea for air-traffic and most of the boat-traffic. The national airline is Air Niugini. This flies to and from Australian cities in codeshare with Qantas, but also to Fiji, Solomon Islands, Singapore, Japan, Hong Kong, Malaysia, and the Philippines. Virgin Blue flies to Brisbane on behalf of the other local airline, Airlines PNG.
Airfares from Australia are quite cheap, especially if you book online and seek one of the special fares. From other countries fares are fairly expensive and it may be cheaper to fly to Cairns and pick up a flight to Port Moresby from there.