Ground zero for hiking in the Baliem Valley is the town of Wamena. You just have to get there to start your trek. Fortunately, Wamena has a brand new airport (as of 2016), which is very civilized, and flying in/out of Wamena is easy. While some blogs and other sources talk about the horrors of trying to get flights, and being delayed for days, etc, that is all bullshit.
From our experience (June 2018), flying in/out of Wamena is just like flying in/out of any airport anywhere. Everything is on-line, everything is booked/paid for in advance, electronic tickets are emailed to you, you show up when you are supposed to, check-in, go through security, and board the plane. Just like anywhere.
We flew from Jakarta on an overnight flight to Jayapura (capital of Papua) via Batik Airlines, arriving at 8 am. I’m pretty sure you must fly to Jayapura first, to get flights to Wamena (but don’t quote me…pretty sure though). We then flew via Trigana Air, at 10:20 am, arriving in Wamena about 11 ish (30 mins flight). Perfect. Everything was seamless and went according to plan.
As far as I can tell, at least three airlines fly Jayapura-Wamena: Trigana Air, Wings Air, and NAM Air. All of them have websites and you can book tickets on-line. We flew Trigana, in a Boeing 737 (OK, it was a bit old, but fine), and it was seamless. And return, on the way out. No issues whatsoever either way.
· No need spend any time in Jayapura (unless you want to for some strange reason). Book a connecting flight to Wamena right after your flight to Jayapura.
· If you have more than a 1-hour stopover in Jayapura airport, go to the Garuda lounge on the second floor. For 80,000 Rp. (5 USD), you have a decent all-you-can-eat buffet, drinks (no alcohol), comfortable chairs, air conditioning, and you can charge your phone. Great deal!