Are there any trans-Pacific flights from the North American countries to the Antipodean ones (Australia, NZ) that would be considered cheap or budget airlines?
The only budget international Pacific airline I'm aware of that is semi-helpful in this regard is Asia Air, but they don't cover this particular entire route - only part, if hypothetically one were to fly to Bangkok first(!).
Essentially after budget airline names for this region to resolve this question.
Most budget airlines rely on short turnaround times for efficiency which you just can't do on long haul flights and fly A320s or Boeing 737s (most budget airlines use aircraft of this class) which top off at 5-6 hour flight duration range. So I'd be surprised to hear of a 'budget' long haul airline which flies trans-Pacific from Vancouver to Bangkok or any South-East Asian / Australia / NZ destination.
However, if you are already in Bangkok and want to fly to Australia / NZ you have a couple of options:
JetStar: Qantas Airways' budget airline arm, this gives you the most number of options from Bangkok to NZ (Auckland and Christchurch) and to a whole bunch of Australian cities. Prices are usually higher on average than budget airlines in the region but they also tend to be more stable. If you are on Oneworld Alliance or Qantas Airlines' frequent flyer programme then you can pay higher rate 'full' fares that allow you to earn frequent flyer points, although you have to consider whether the amount of points you earn is worth the price difference from 'lite' fares.
Tiger Airways: From Bangkok they only fly to Perth, which may not be what you want. Their hub is in Singapore and they used to have flights from Singapore to NZ but I don't see that option any more.
Air Asia: Air Asia's hub is Kuala Lumpur, so if you fly with them you will have to fly via KUL. They connect to Christchurch in NZ, and Australia (Melbourne, Perth, Gold Coast).
For general flight cost comparison in this region I would suggest using Wego.com which is a Singapore-based travel aggregator much like Kayak but often throws up options other travel sites miss out.