Yes - we call them mobile phones here, which ends up being just "mobile". You can get them at any of the phone companies and also at the local post offices. Here is some detailed info given on a backpacker's site. http://www.bugaustralia.com/essentials/mobile-phones-for-travellers-in-australia.html

I think you will need a different one for NZ than Australia unless you want to pay international rates in one country or the other. In NZ if you phone takes a SIM card just buy either a pre pay 2 degrees SIM or vodafone SIM and put it in your phone. Both are super cheap to purchase. I would recommend 2 degrees but you may have to check into what internatational calls cost.

I use www.amaysim.com.au for best value local and international calls. I have 4 sims I use for business and private use. A phone can be bought for under $A50 at Big W, Coles, Woolworths, Dick Smith or Australia Post outlets.

Thank you for the comments
Z

Agree about Australia Post - a range of phones. If you browse this Australia Post catalogue there are some phones at the end:
http://catalogue.auspost.com.au/portal/

As another cheap sim alternative try Lebara - excellent rates if you want to call overseas and they will send you a free sim so you have one before leaving home: https://shop.lebara-mobile.com.au:8081/.
From discussions on the Europe forums Lebara may not send it to you in another country unless you sneak in 'USA' in one of the other fields; not sure about Lebara in Australia (they are international). The discussions mostly revolve around France.

Maybe look at buying an unlocked quad band phone if you want to subsequently use it in many other countries. Possibly a little more than any old phone (mine is a cheapie but not the absolute cheapest, and cost me about $120 from Dick Smith) but you will extend its life considerably as you can then use it in Europe and the US.

Enjoy your trip!

Lavandula

This message maybe too late but I bought a Telstra prepaid card for $30 which was perfect for 4 weeks. My phone with T-Mobil had been unlocked before my trip to Australia and I was very happy with the card.Don't know how New Zealand works since it is a different country.

I was told that Vodaphone had the worst reception.

In Australia, the Telstra network has the best coverage with Optus next and Vodaphone third.

I also use Amaysim as their call rates are much lower than say, Optus or Telstra. Amaysim uses the Optus network. SIM cards are $2 from supermarkets, Dick Smith, JB, etc.