I use kayak.com (or their iPhone App) to find out the routing and my preferred flights and then either book online or go through a travel agent.
For the long haul, yes, choose according to your preferred FQTV program. And it will probably be worth to buy a return ticket. You would for instance fly to SIN and return and to the other legs from SIN. It will add one leg but would most probably reduce a lot your total cost. And you might want to consider flying East (via Europe): slightly longer but one stop instead of minimum 2 from MIA.
Agree furthermore with GeoMedic: do consider also RTW tickets.
And in your case, I think I would go through an international travel agent as the trip is actually quite complex (international because you want to have a help number in Asia).