Methods/ routes of transportation avoiding Port Moresby

Any suggestions on how to move around Melanesia's nation states without stopping through Port Moresby? Specifically, I need to travel quickly and frequently between Vanuatu, the Solomons, Fiji, and Bougainville.

Coming from the east, you can't really avoid it, other than simply flying out of the airport to someplace else in PNG right after flying in internationally.

In any case, I met several tourists who (after hearing all the stories) were pleasantly surprised by how "beautiful and friendly" Moresby was - something I am yet to see for myself, as I always entered the country at its opposite end, from Indonesia.

Except for Bougainville, you can travel between all the others without even thinking about PNG. Both Solomon Airlines and Air Pacific have direct flights between Fiji, Vanuatu and Solomon Islands, without going near PNG, which is in the other direction.
Solomon Airlines have 2 flights a week in each direction, and go from Honiara to Santo and Nadi, Air Pacific go once a week Nadi - Vila - Honiara and back.
To travel from any of these countries to Bougainville, you must either fly via Port Moresby or go by open boat between the Western Solomons and Bougainville, which may or may not be legal.

Don't worry if you have a stopover in Port Moresby.

Hotels in Moresby are generally well guarded. They might be able to pick you up from the airport. Taxi's are also available and safe.

When I was in Moresby in 2003, I walked around and travelled by bus during daytime without problems - both in town and around Gordons. Just don't take your valuables or at least don't show them off. People were indeed friendly: always happy to give me directions or have a little chat (people don't do that here in Amsterdam).