"I'm at the airport waiting to get on the plane"
 

"I'm waiting at the airport to get on the plane"
The literal translation is "Estoy en el aeropuerto esperando subir al avion".

I'm wondering if there is a more commonly used way to say it, or if this is pretty much how people all over Latin America say it.

I think that's what you're most likely to hear... probable variations would include "para subir" or "para poder subir", but the meaning is the same.

If memory serves, the word abordar,abordamiento is used instead of subir.

Oh, very true, Poli. "Estoy esperando para abordar (el avión)" is also very common. :)