Go Steph! You won't regret it!
I was in a pretty similar situation to you except that I owned even less than you if you can believe it!
I started out by finding a flat. This is 5 years ago now, but it cost me $250 per week for a two bedroom flat in Mt Albert (close to uni) and it was really secure which was a biggie for me being on my own. The bond was $750 plus a letting fee, WINZ lent me the money for the bond. From memory, my weekly benefit including accomodation supplement worked out at just over $500 a week so after I paid the rent I had $250-odd to pay the bills and groceries etc. That was pre-WFF tho so it should be more these days. In my experience, you're better to rent thru an agency, yes you have to pay a letting fee, but the agency will make sure any repairs etc. get done promptly, whereas if you have a private landlord there can be a lot more mucking around (and if they're anything like mine they'll do anything to cut costs

)
I second what someone else said about the kitchen starter set, I got mine at the WareWhare for about $100, yes it's pretty crappy stuff but it included 3 pots, kitchen utensils, 2 frying pans etc. I also got a dinner set starter kit which was about $60 for dinner plates, bread and butter plates, bowls, cutlery, mugs and glasses, so all up a cheap way to kit out a kitchen. Amazingly enough I *still* have the pots, most of the cutlery and random bits of the dinner set that haven't been broken and they are still going strong.
Whiteware - I bought a washing machine, dryer and fridge from the T and E, TM would be the modern option tho lol. WINZ will give you a grant to get them but only if you buy from a store and only second hand. They expect you to get 3 quotes for each item and they'll pay the cheapest. Unlike someone else said tho, you DO have to pay this back, it's called a Recoverable Assistance Grant (as opposed to a Special Needs Grant/ Non-recoverable assistance grant for food or emergency medical/dental care). I found it worked better for me to borrow $1000 off my mum, I managed to make that cover washer, dryer and fridge plus all the kitchen stuff whereas if I'd had to used WINZ grants and buy from a store I would have ended up paying more for my whiteware.
When it came to actually moving, I did it all in one day. I hired a trailer and we just moved load after load till we had it all in there. I didn't have much furniture but I'm a hoarder (yes, I have OCD!) so I had boxes and boxes of just stuff.
Lounge suite I got from a friend of a friend, it was crappy quality second hand but for $200 it was fine with a cover over it and lasted till we moved in here 4 years later and could afford to buy a decent one. I'd definitely take Fiona up on her offer re: the lounge suite coz they are expensive.
I didn't even have a TV etc, my uncle gave me a tiny 12" which now lives in our bedroom, I replaced it with a decent one once I had enough $$$. In fact, that's pretty much what I did with everything, as I had more money I bought new things to replace the second hand ones, I now have a new washer, dryer and fridge plus have bought a spare queen and single bed etc.
Like I said, best of luck! I'm happy to help any way I can, and if you need stuff like linen I can have a dig around here, unfortunately having a Willie that works at the tip I don't hang on to surplus furniture etc. so don't have any of that. But go you! You will not regret it! I was on my own with Maya for 18 months and it was the making of me, I learnt to be independent, to budget, to cook decent meals, even to do the 'man' stuff like put the rubbish out and mow the lawns.
Oh and one more thing, WINZ will also help out with moving costs if need be (again you have to pay it back) and clothes for Lucas if things get tight. And do use your $300 a year food grant allowance, every little bit helps. They'll probably only give it to you in $75-$100 lots but it comes in handy if you have a week where the car breaks down or Lucas gets sick and you have to pay doctors fees etc.
Re: the car - do you have to sell it or can you work out an arrangement with your Dad to pay off the balance at a manageable weekly amount?