As the title implies, whenever I click "game" on the menu at the top, I just get a white screen - I have screencapped this, and here is a link : http://gyazo.com/b2706c04f1a3aa9ead018cec90e40beb.png
What am I doing wrong?
Had the same issue, updating my Java fixed it.
Like Vanguard said, It is most likely the case that your Java is outdated. Here is a link to a newly updated java, Java Download (http://www.oracle.com/technetwork/java/javase/downloads/jdk7-downloads-1880260.html)
Use the downloadable client it's much better, it may be a browser based problem.