I spent a year in New Zealand in 2002/2003. Awesome country!
I spent the whole time hitchhiking and found it easier to get around hitching in NZ than anywhere else I've travelled. Even got picked up by a Rolls Royce limo! (
https://www.facebook.com/photo.php?fbid=18643541212&set=a.8433446212.24975.516031212&type=3&theater )
It's a great way to really explore a country, especially if you want to split off from your family for a week or 2 for an adventure with a mate!

It would be a lot harder with a group of 6! You could split up into 3 pairs and meet up at a set destination though.
I found it easier hitching with a girl, but a lot harder if it was me and another guy.
If you do have time to get to the South Island, I loved some of their multi day hikes! I think they called it tramping...
My favourite ones were:
Trekking from Lewis Pass to Arthurs Pass (6 days)
St James' Walkway (5 days if you want to go slow)
And one which tied a few routes together. I set off from Wanaka towards Mt Aspiring and I emerged 6 days later near Queenstown.
Spent a month in the Coromandel too. That was cool, but it gets quite touristy in the summer.
I wouldn't read too many tourist guides (Lonely Planet etc.) unless you want to find yourself surrounded by tourists!
I asked almost every Kiwi I met what their favourite thing was in NZ and I had an endless supply of destinations and quests to discover and explore just from their advice!