Posted too early.....
Agencies are the way to go for temp Government dept jobs. I did try the big global names, Hays etc, but found the NZ ones better. I only ever got my work through one person (2 agencies, as she moved jobs while I was there), and think that finding someone I got on with helped! Beyond was one of them, can't remember the name of the other.... They won't be able to help you until you're there and have your visa to show them.
They will test your typing skills, so brush up before you go! I got paid about $25-30/hour for not very difficult admin work, which at the time was about £12-15/hour - way better than here for what I was doing but cost of living is higher - quality of life much more fun, though!
What else? It rains a lot in Wellington, and is v v v windy! Everyone will comment on the weather when it isn't windy! Forget taking an umbrella, no point, will end up in the bin after a day, long-ish coat with hood is required (get here, way too expensive there). And they do get earthquakes - often. Mostly small tremors that you'll get used to. I was there for a big one in Wellington in 2013, but you just learn to deal with it.
Cinema is big there, lots of lovely, quirky indie places with a bar, so glass of wine + movie is the norm.
I probably have loads more 'advice' but it's more exciting to find out for yourself. I really miss the place once I start talking about it, and can still picture the drive around the coast road, on the look out for orcas in the bay (never saw any, or any 'penguins crossing' despite the road signs....)