Hi onedog, what we talk about all the time.....Weather.
Sydney will be on the mild to chilly side, moist but not cold.
Cairns will be warm dry and pleasant.
Don't worry about an umbrella, if you feel you need one, department stores sell them cheaply.
Pack a sweater for Sydney and shorts for Cairns
New Zealand is beautiful and their dollar and ours is down at the moment so you couldn't pick a better time to BUY your tickets and a wad of the Australian currency. My advice is to pre pay as much as you can and remember that fuel taxes are being lifted and $250 Australian equates to $160 US, very good buying. The dollar will start to rise in February and should be healthy again by this time next year. Enjoy the trip, if you enjoy train travel, the BNE to CNS run is superb, and Brissie is a warm friendly spot to stay a day or two, regards bkkrop