As with most places, it depends on what time of the year you visit the world's southernmost capital city.
Wellington is known for its brisk winds (it's not called Windy Wellington for nothing) and it's been said that only fools and tourists unfurl an umbrella here! But in winter, the temperature seldom slips below 10 degrees Celsius and in my lifetime, there's only been snow a few times. So if you're here in autumn or winter, make sure you bring a good coat with a hood. And gloves and a scarf. And, of course, comfortable walking shoes.
There are plenty of shops to stock up on beautiful New Zealand merino knit clothing so if you like to travel light, you can buy what you need here. In New Zealand, we run 240v electricity; but your travel agent and/or hotel should be able to help with conversions.
In summer, it can get quite hot and the Ozone layer is apparently less robust in the Southern Hemisphere, meaning the sun can burn more quickly. Bring a good sunscreen and a sunhat and make sure you use both! The local mantra that all Kiwis learn from a young age, is Slip, Slop, Slap (slip on a shirt, slop on sunscreen and slap on a hat). So do as the locals do.