Australia Bike Rental

Personally, for 10 days i wouldn't fool with taking a bike all the way down there.  I'm not sure about any rentals but you may see if you can find a local triathlete club down there ( depending on which city ) and see if you can get in touch with anyone to let you borrow a bike / ride with them if they have multiple bikes.

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Hey. Pack your warm riding gear. It's a little fresh at the moment. I would also agree with checking out clubs or tri shops as they will also be able to help you with suggested training ride routes.
If you're coming to Sydney, check out Cheeky Monkey (http://www.cheekymonkeymultisport.com.au/). Not sure about rentals. They may rent out bikes but they're very switched on there and should be able to suggest where to perhaps get a bike if they can't help. One of the bigger clubs is BRAT (http://www.bratclub.com.au/). If you get intouch with some of those guys you may also get some help and find some people to ride with.
If you are going to Melbourne check out CBD cycles which are massive (http://www.cbdcycles.com.au). I am not from Melbourne but there seems to be a massive cycling scene there.
Not sure what else to suggest in Melbourne and I am sure there are similar options in Brisbane and Perth if that is where you are headed. If you are headed to Sydney though I would definitely start at Cheeky Monkey.

Good luck. Enjoy your trip!! Nice place.

One of the best swimming cities in the world. There's three (plus another two in summer) 50m pools within the CBD and environs.

Not sure that Cheeky Monkey do rentals (give them a call on 0-11-61-293993370) but there's another place a couple of doors up from them that does but think they're only cruisers for the park etc.

Actually bringing your bike here isn't so bad. I travel with mine as soon as I know I'm going to be away more than a week (travel to London, NYC, Hong Kong and Singapore a lot - though I would never take it to HK). Take about 15min to pack and unpack with practice and I've never had issues. Once you're on the plane it doesn't matter whether its a 60min flight or a 20hr flight.