Just about any school in NZ can sort you out a student visa, we haven't gone down the road of the USA yet (in fact the school where I work is picking up quite a few enquiries from UK/Europe now that the US training option is such a big pain in the @rse.
Training's generally cheap, expect to pay between $120 and $150 NZ per hour for single and about $300 - $350 for twin training (divide by 3 to get the equivalent in pounds).
As Anti Skid points out, you can get a government funded student loan to pay for all your licences & ratings if you're a NZ national, unlikely to apply to you but what it does mean is that every year the schools are banging out another few hundred wannabes with taxpayer funded CPL/IRs, in a country of only 3 million people you can see how the problem arises, I read somewhere that there are 12 unemployed CPL holders for every paid flying position in this country.
(Hence why most of us end up flying for low cost airlines in the UK instead
)
Theres loads and loads of good schools in NZ, you can't really go wrong unless you train at one of the Ardmore based schools (big overpriced congested-airfield sausage factory nightmare) or Massey School of Aviation (the laughing stock of the NZ industry).
I see you're in St Andrews divorcingjack, strongly suggest you should take a quick drive across the river to Tayside Aviation in Dundee and ask to have a chat to the New Zealand instructor that works there, he's an absolutely a top bloke and will give you loads and loads of sage advice (for the small price of a pint in Laings bar). Or feel free to email me if you want.