For SE Asia and Europe (mainly Greece) you would be best off with a card that allows the points to be transferred to SQ KrisFlyer although I don't think SQ flies to DRW. The reason I mention KrisFlyer is that it requires less points for awards than Qantas.
A couple of exaples are for economy SYD/MEL/BNE-SIN/BKK/HKG requires 60,000 QFF points but only 46,750 KrisFlyer miles (actually 55,000 KrisFlyer miles - 15% online booking discount) and SYD/MEL/BNE-ATH requires 128,000 QFF points with transit in SIN/BKK/HKG and LHR but only 80,750 KrisFlyer miles (actually 95,000 KrisFlyer miles - 15% online booking discount) with transit in SIN only.
Taxes and surcharges are roughly the same for both airlines. You may also find that SQ KrisFlyer comes out better for awards in premium cabins.
Which card? Some type of Amex credit card (you choose the colour and obviously Platinum would cost more but also earn more points) with Membership Rewards or a direct sweep Amex like the Westpac KrisFlyer Amex credit card.
If your heart is set on directing points to QFF then you will need some type of co-branded Amex such as CBA, ANZ, Earth etc.
Don't know much about Visa/Mastercard.