Yes, NZ born citizen have automatic entry into Australia (not sure about migrants). You're a legal resident, but not a citizen so you can't vote (bummer! lol)
Got my job through recruitment agencies, as have almost everyone I know. Easy as, saves trolling papers and net sites, they call you with jobs akin to what you want. My girl and I both had full time jobs, and a flat, within a week of getting here! But that was pretty jammy really.
As for websites, seek.com.au is the biggest and gets pages and pages and pages of new jobs everyday!
Living in Melbourne, fucking loving it! Climate loads better than Wellington, but colder than Auckland in winter months. Hot in summer, but a dry heat generally, none of this 85% humidity muggy as fuck sweating through your undies while doing absolutely nothing crap! More parties than one can shake a stick at, cool gigs on almost every weekend for whatever sounds you're into really and 25000-30000 festivals spread across 7 or more stages in the summer. No conplaints, no regrets!