The best bet for you to get miles fairly quickly and be able to redeem them to go to Australia is through Citi/AA cards. You can get two Citi/AA cards at one time (the Visa and the Amex), both of which give you 50k AA miles. After 65 days, you could get the business version of the visa card and be at 150k without even putting anything other than the minimum spend on them.
For a roundtrip ticket to Australia from Washington to it is 75k in economy. With those 3 cards, you'd already have enough AA miles to fly there.
The other option would be to go the Chase route, as Chase points transfer to United at a 1:1 ratio. You'd need 160k in Chase points to make the flights for 2 people in economy, and could rack this up pretty quickly if you got the Chase Sapphire (50k), Chase Ink Bold Business (50k) and the Chase United card (50k-targeted).
I like the Chase Sapphire the best for everyday spend, but the AA option allows you to get all the miles you need at one time and the minimum spend on the cards is less than the Chase cards.
Either way, you can't go wrong, and those free flights are easily within reach.