Qantas usually do lots of different deals on flights, a friend flew from the Uk to Oz and I think the fare he used was called Dream Time Plus which allowed for one stopover each way to and from and two internal flights within Australia, but it was about £150 more than the normal fare.
I think that if you are only going to one other place you would be much better to use one of the cheap Australian flight companies Virgin Blue or Jetstar who do really cheap internal flights and as Melbourne - Sydney is not too far you could probably get a round trip ticket for much less than £150. Another option is to fly into Melbourne from the UK stay with your Aunt then fly back to the UK from Sydney, then you would have to only pay for a one way ticket from Melbourne to Sydney. Most airlines allow you to fly into one City and leave from another. I flew in to Adelaide and left from Brisbane.
I had to fly internal flights with Qantas because the cheap airlines did not fly the routes that I wanted within Australia and the flights cost me £400, had I flown from the Uk with Quantas these flights would only have cost me approx. £150.
Unfortunately the 20Kg weight limit is pretty much standard these days.