I am wondering if people could recommend things to do and see in Australia and New Zealand for one month because with such a short trip, I do not want to waste it visiting places that would not be worth it
See here for Oz. (Sorry, crossed posts with Shawady)
For NZ, Wellington is a great city, Rotorua is a good base for activities. Mount Doom from Lord of the Rings (apologies, can't remember the right name) is a brilliant hike.
IMO, South Island is the better. We hired a car and drove down the west coast, absolutely breath-taking. Queenstown is beautiful, as is Milford sound.
Take a ten days to two week drive around the south island of NZ, don't miss Nelson, the West Coast incl Milford Sound, Queenstown and the lakes, Fox Glacier, and whale watching off Kaikoura (spellng from memory!). The Alps are spectacular, if you get a chance to land on the glacier high up in the mountains on the ski plane, you'll never forget the experience.
Most important thing to bring to NZ? A good camera, it's the most photogenic country in the world IMO.
I suggest you visit South Island in New Zealand, its very beautiful, Milford/Doubtful Sound are fijords in the South Island the scenery is incredible. If you have any interest in outdoor activities you should visit Queenstown, there is white water rafting, bungee jumping and skiing to do. There is a small town called Kaikura were you can go whale watching and swim with dolphins its a great experience. Christchurch is a pretty city if a little quiet and provincial. Finally there is a train jouney named the Trans Alpine Express
With such little time, internal flights in oz will have to be your way me thinks. Your probably flying into Sydney or Melbourne to begin with and will need a few days to climatise which gives you a chance to experience one of the two big cities anyway. To go to Oz and not visit Ayers Rock would be a pity and is highly recommended..a must see! Book an Internal flight to Alice Springs and you can do a few days touring in the outback to the rock..you wont be disappointed!. Next you could fly up to Cairns and make your way down the east coast. A place worth visiting is Arylie Beach from where you can do the Whitsundays a must see in OZ. You should also consider Fraizer Island which is a two night camping trip and is another highlight of the oz experience. Noosa is another town on the east coast you will probably encounter and is well worth the visit. Try and make the Australian zoo (Steve Irwin) in Brisbane great day out. All of the above should be managable within two weeks. (Best advice is to hire a camping van and drive down the east coast to really enjoy the experience). This will still give you two weeks in NZ but as another post said the south island is better and with your tight schedule you'd be better to stick to here. Queenstown, Milford sounds and Franz Joseph glaciers are amazing places and should not be missed. If its the right time of year you will also be able to get in some skiing. Where ever you go you will be guaranteed to have the time of your life. Enjoy your trip!