We came from the UK last December. We looked for a removal company about 6 months in advance, choosing a shortlist of 3 companies headquartered in our locality (south west London). They each visited and quoted a price. We made our decision not only based on the price, but on overall impressions of their customer service, what was included in the quote, what services were offered at the other (NZ) side. To be honest there wasn't too much difference and they all sounded perfectly good. We had our packing done by the company as we've found that this is by far the quickest and best way - these guys are really professional and will pack up your valuables really well and really quickly.
NZ is really strict on cleanliness, in an effort to keep out nasty bugs and insects. We had to sign a declaration of cleanliness, and in order to sign it truthfully we had to buy an approved detergent and clean our shoes with it, plus my husband's work tools etc - anything that could be contaminated with earth. There are restrictions on what you can bring into the country so it's essential to check these before you pack.
Everything went fairly smoothly, with the main issues being at the NZ side (it took about a month from the moment our goods arrived in NZ to actually get them delivered to our door, partly due to communication issues with the NZ partner company and partly due to customs clearance times. If we did it again we wouldn't bother bringing any electrical appliances such as TV, DVD, printer etc because they don't work well here and in the case of our printer, we found that the toner cartridges we use aren't available in NZ! Most things here are more expensive in the UK, not sure if this is also true of the US.