If you are concerned about a few hundred dollars Australia may not be the place for you. Due to exchange rate & distances in Aust it will be very expensive.

No airlines flys ORD - SYD or Mexico to Australia (without a transit/stopover)
Airlines to try include Virgin, Air NZ, UA, Delta, Air Canada [?], Hawaian & Qantas
If you are heading north flying into Brisbane [BNE] may be a better option. SYD is not the only international airport in Australia .

Airlines within Aust are Virgin, Qantas & Jetstar
If you are backpacking sharing lifts from a hostel often works

Looks like you need to do a lot more research. Looks at other threads here
Also http://www.lonelyplanet.com/thorntree/index.jspa a good place to start for backpacking type travel.