There's a MUCH easier way to figre out the possible routings than starting with the Wikipedia page for the Sydney airport.
Use Award Hacker
. Specifically, for your route http://www.awardhacker.com/#f=Chicago&t=Sydney&o=0&c=y&s=1&p=0
This tool will tell you which miles to use and which program you can transfer points from. The cheapest result is 60k JL (that's Japan airlines) miles, redeemed for a flight that's actually operated by AA (American airlines). AFAIK there's no viable JL cc available in the US, so the best way to get their points would be to transfer them from SPG (Starwood Preferred Guest). Unfortunately, the SPG cc signup bonus has never exceed 35k points AFAIK, so if you don't already have a bunch of SPG points, that's out.
The first viable option is 70K NH miles (That's Al Nippon Airways) on a flight operated by NH. It does involve a transfer in Japan (I assume Tokyo). You can transfer Amex Membership Rewards (MR) points there. You'll need 70k of those, which usually isn't available from just one card, but sometimes is. I've seen a 75k MR points bonus on the Business Gold Rewards card (need a business - work reimbursed travel expenses DO NOT count). I've also seen 100k Amex MR points in the past from the various flavors of the Amex Plat, but I don't think they're available at the moment. So it'll probably require two or more cards - 50k from an Amex Premier Rewards Gold (PRG) and 25k from an Amex Green (not always available). There's also 10k from an Amex Everyday.
The sad truth is if you're starting from zero, barring a 75k or 100k Amex MR signup bonus on the BGR or Plat, respectively, you're going to need at least two cards no matter what routing you pick because no other single card offers more than 50k, or sometimes 60k, miles as a signup bonus.
Now with that all said, Award Hacker only checks what is possible via the mileage charts. It does NOT check if that routing is available on the dates you wish to fly - in fact, it doesn't even ask what dates. Once you find an option from this site, you need to check if such an award seat is available by checking the appropriate airline's website for a ticket booked wih miles.