How to obtain and maintain your violin
Buying violins (in NZ)
The easiest "click a button and it arrives at your door" option, Endeavour "Optimist" violins from the String Workshop.
These violins are not the cheapest ($NZ 280 delivered at the time of writing this) but they are the cheapest I have found that actually work. Many other student-grade violins that I have tried have had problems very early on that were difficult to fix. Any occasionally issues I've had with the String Workshop violins, usually many months after purchase, have been fixed promptly and for free.
String Workshop has a range of violin qualities from the "Optimist" model up (though I have no experience of his upper-range violins at ~$NZ 10,000-15,000). The other String Workshop model I have experience of is the "Journeyman" model violins (~$NZ 480 at the time of writing), which I have used when students want to upgrade a little from the "Optimist" but still at an affordable price.
If you can spend more then I highly recommend Antonio Strings.
The Antonio Strings AS10 model is their cheapest option (at ~$NZ 900) and it comes in all sizes.
If you go to their Violins page you'll find a range of violin models from this one up.
Hiring violins (in NZ)
In other areas of New Zealand (and in the Nelson area) music shops sometimes hire out violins too.
What size violin to get for your child
Buying shoulder rests
I recommend that very small children's violins (size 1/16 and 1/10) just use a foam.
For children's violins larger than 1/10 I recommend this shoulder rest.
For larger children's violins (3/4 size) and full size violins, I suggest either
From the String Workshop: Everest shoulder rests
From Antonio Strings: Kun Original shoulder rest (scroll down that page).
Strings come in different sizes for different sized violins. You can buy strings individually or as a set of 4. Even within a set, strings vary considerably in price with E-strings being the cheapest and G-strings being the most expensive.
For children's violins I suggest Vision strings (which can be ordered by email from Antonio Strings).
For full size violins, you might want to try various different strings to see which sound you like best. Vision strings are popular, along with Dominant strings. Both can be ordered by email from Antonio Strings.