As an Australian citizen living in NZ, you can sponsor your partner and family for Residence in NZ. This page tells you about the forms you will need, and that your partner will need.
http://www.immigration.govt.nz/migra...ly/default.htm (He will be the applicant, for himself and the children, and you will be the sponsor.) Look at F2.10.5 a and b here,
http://www.immigration.govt.nz/opsmanual/46176.htm, and you'll see that you are going to be able to do this, but also see section e. Someone else asked Immigration NZ about this situation - it had looked to me from this regulation that he would have to wait until he was in NZ to apply, but they told him he could start the process from outside NZ.
http://www.enz.org/forum/showthread.php?t=40660 It looks as if it depends on the individual's own circumstances, so you should ask INZ.
If you're not able to apply before you move, your partner and children could travel on visitor's visas, and then you could start the process from within NZ.
About your partner being able to work as soon as he arrives in NZ - that will depend on whether INZ allow him to put in for the visas for himself and the children (sponsored by you) before arriving in NZ. If they don't, and if they have to travel on visitors' visas, he will not be allowed to work immediately. However, partner-sponsored visas are normally processed relatively quickly - supplying plenty of well-organized evidence helps the process. (Old threads here on the subject of evidence of partnership.
https://www.google.co.uk/#q=site:enz...tnership+proof)