Storm Glider,
As an Australian, the US is similar in the ways of building hours and going up the ladder. Australia and the US have a good sized General Aviation sector.
Europe and the Middle East "basically" have flight schools then straight to airliners (737-A320). This is only available to citizens from cadetships or self sponsored flight schools. It is very competitive, even to a point of a lot of qualified pilots not gaining a job for a year or two even three.
Best to start in the US and build your hours and experience. If you can go reserve in the USAF just to keep things afloat will be one option?
Because the first job or two will pay for cans of baked beans and toast.
Forget NZ, as it is a very small market and a lot of green pilots make the hop over to Australia to gain experience (Australia and NZ have a trans tasman agreement).
With regards to working in Australia, there is a visa called a 457, however you will need 3000hrs plus and a ATP to apply with some time on a twin turbo prop at 12,500 pounds and above. There is only 2 or 3 regional airlines that will support this. But hey, lets focus on the first step in the US.
So basically, train in the US and you have a bigger aviation sector to choose from.
Hope this helps?