Using the new system is easy enough; simply visit the Contact Blizzard Support page, select the game you need assistance with, and then choose from a variety of dropdown menus to pick the selections that best address your particular issue. You'll see three buttons at the bottom of the screen, one for submitting a ticket, one for contacting by phone, and one for live Web Chat. Please note that web chat and phone support availability still varies based on hours of operation, as well as the type of issue you need addressed.
Honestly, I'm wondering why this type of support system wasn't put into place years ago. With the Web Chat feature, players can bypass sitting on the phone and listening to hold music, and just chat with a representative to get their issues addressed. And with the new filtering system for issues, it makes it a lot easier to figure out what I should be doing if I have a problem. Hey, don't get me wrong, I do love Blizzard, but if I don't have to sit in a phone queue, I'm generally all the happier for it.
You can read more about the new live Web Chat feature on the official site.