Most travellers arrived and depart from West Papua by air through the popular gateways of Sorong, Manokwari, Jayapura (Sentani) or Biak. Fakfak on the Bomberai Peninsular and Timika also have airports. The only direct link is between Jakarta and Jayapura with Garuda and Lion; all other routes are serviced by other regional carriers and generally go via Denpasar and Makassar. You can find and book flights through tiket.com. Note that Jayapura doesn’t actually have its own airport; it’s serviced by Sentani Airport 25km to the west.
A land border crossings exists between Jayapura,and Vanimo, Papua New Guinea for those prepared to make the difficult overland journey. The crossing is notorious for closing without notice so it’s unreliable at best. Nor does it have Free or Visa on Arrival facilities so you’ll need to arrange your visa in advance.
Pelni ships operate regular passenger services between Java and Papua, stopping at various islands and ports along the way. The entire journey takes around 7 days, less obviously if you’ve caught the boat from somewhere closer. Along the north coast, the Pelni shipping route is either Sorong, Manokwari, Biak then Jayapura or Sorong, Manokwari, Nabire, Serui, Jayapura. Pelni ships service the south coast but less frequently, stopping at Fakfak, Kainmana, Timika, Agats and Merauke. You can check schedule and prices on the Pelni website using Google Translate.
Perintis (cargo) ships operate to neighbouring islands and many of Papua’s smaller ports but they usually only run when they have a full load. You might get lucky and find one heading in the direction you want, when you want if you ask around the port but it’s unlikely.