Depends what you like... In Feb and March I'd stay away from anything North of Sydney. Queensland is 30degC plus with 98% humidity. Basically like Florida in July. Also, as Queenslanders haven't figured out how to change their clocks they do not have daylight savings time and it gets dark early... May to September is OK in Queensland and thats about it. Just my opinion.
I would spend three or four days in Sydney. It is crowded but the rocks and the harbour are outstanding. If you are going to rent a bike, Melbourne and Adelaide have some great Mountain and Coastal Rides. It does get warm but very dry so its a bit more comfortable. Adelaide has a lot of wineries in the hills and the beaches are also pretty good.
You can rent most bikes including Triumphs see
http://www.garnersmotorcycles.com.au/gm_rates.html You may want to consider a trip to Ayers Rock and Alice Springs. It will be very hot 38C, dry and tons of flies BUT this is the real Australian Bush. Wouldn't spend more than four days there but I think its worth it.
New Zealand:
Would stick to the South Island. Blenheim, Nelson, Christchurch, Queenstown/Wanaka and Milford Sound. That time of year would be ideal for a bike. Campervan rentals are very popular and a good way to tour the South Island. My folks come out from the States and did a South Island Trip in two weeks with a Campervan and it was rushed but they loved it.
