Consider attending conferences as well, especially as a volunteer. Aside from getting access without paying the costs (which can be pretty steep for something like GDC) it is great for networking.
The volunteer application for GDC Eu in August is open for another week, and while it is smaller than the show in San Francisco, it has it's benefits. Ask me if you'd like me details on that (just don't take my spot as a CA ;) )
There are other conferences going on all the time all over the world, including New Zealand to check out.
I'm not a pro by any means, but I've learned a lot trying to break in (as a concept artist coming from Israel, I knew the odds were stacked against me.)
For instance, Europe has a good number of game studios, a lot of whom speak English, since employing people from all over the world is common practice.
Japan and South Korea on the other hand have much bigger language barriers (but in Korea studios list addresses, and a lot are super nice when you give them a surprise visit, toting your portfolio. The locals don't do that kind of thing )
You shouldn't limit yourself to AAA studios. It would be the way to go if you wanted to move to the US, as they would be better prepared for immigration issues, but you can get just as much valuable experience at an indie, mobile or outsourcing studio.