Yep, depends on what's harder for you. In my case I didn't have my birth certificate here, and it was a real pain to get it while already in NZ. You could also bring letters from employers on a letterhead with the company's logo. Bring some payslips from the previous work experience also.
If you have to prove partnership, papers like what Jandm said, marriage certificates, joint bank account statements, etc....
Police certs expire after 6 months, so unless you are going to apply just after arriving, I would say you probably want to do it from here (not sure if Canada has an embassy in NZ or not).
If you have to prove your english, from attending studies in english or something like that then bring the original certificates or transcripts that states that they were done in english.
If you need to get your qualifications assessed by NZQA bring not only the diploma but also the transcript with the curricula and plan of each course. Not sure if Canada is part of it, but there is something called Apostille of the Haye convention or something like that, which will be given by your country of residence to prove that your Diploma is not fake and is valid in the country.
I think that's mostly it. But from my experience if you are in doubt about bringing it, then just bring it.