My spouse and I are going for 3-weeks vacation on the southern island of New Zealand. We intend to go for one multi-day trek (the Kepler Track), and a few more day treks.
Do we need water-resistant (e.g. Goretex) shoes for the trip?
I am a big fan of the low ankle water resistant Salomon fell-running shoes.
The Gore-Tex allows moisture and sweat to escape rapidly, and when worn with wicking socks they actually work well to keep your feet dry and sweat-free.
I do most of my hiking in Scotland, which is on a par with New Zealand for precipitation - I would definitely recommend water resistant fell runners, along with multiple pairs of socks. You will be much more comfortable, will have less chafing and blisters and generally be much more comfortable.
Think about whether it really matters if your feet stay dry. Will this be in warm enough weather when trenchfoot and frostbite are not problems? If so, the simplest answer may be to let the feet get wet. Wet or damp feet by themselves is not really a problem.
I do most of my hiking in New England where it can rain any time on short notice, and soggy ground and stream crossings are common. I realized a long time ago that it is simpler, less stressful, and actually more comfortable in the long run to simply accept the fact that water is going to get into the shoes at some point during a hike in warm weather. It is simpler because ordinary running shoes are fine. Less stressfull because you stop worrying about tip-toeing around puddles and how to get accross streams without getting wet. It is more comfotable in the long run because the smaller and lighter shoes outweighs the feeling of wet feet that some may find bothersome. The latter is simply a mindset issue, so is something under your own control.
If you have shoes that let water in, they will also let water out readily. If your feet can breath, then wetness by itself is really not a problem - again assuming warm conditions. If you are going on a longer hike, bring some extra socks and change them frequently. You can have a couple of pairs of socks drying while you are wearing a third, for example.
I avoid water-resistant shoes for hiking since they trap moisture (your sweat) in as well as out. Kinda depends on hiking style, too - if you're ultralighting then you'd probably want trail-running shoes that breath and dry quickly as you walk. See Ray J.'s advice. They're light enough that people often carry a second pair.
Another factor is weather - a bit of rain is no big deal, but if you're tramping through snow, then I'd think you'd want some water-resistant regardless of your style.
I personally wear Chaco RedRock shoes. They're heavier than running shoes (about 16oz each), but they hold up well and have the high arches I need. Unfortunately, they take too long to dry, but it's usually not a huge deal.
If you're conventional heavy-pack you'll probably want ankle support, which means boots.
Whatever you get, do some trail walking in them before you go. Boots need break-in, and hiking in shoes means you want to strengthen your ankles before you go. Good arch support that matches your foot will help avoid ankle sprains.
When hiking for multiple days you will likely encounter scenarios that you will get wet feet regardless of the footwear you are using. My experience with Goretex footwear is the same as most fellow hikers I know: Goretex footwear can delay your feet getting wet but it cannot prevent it altogether. The important question is: What is your strategy once your footwear is wet?
Having your feet wet for extended periods of time can become a serious problem on the trail. In my experience, Goretex shoes do not help in avoiding those situations. The drying time of Goretex footwear is usually higher then quick dry shoes. Because of their long drying time quick drying shoes are often preferred for long treks.
In the past I have accelerated the drying of my Goretex footwear by using pack towels to accelerate the drying process once it stops raining for example. If I`m lucky enough to hit a town then stuffing news paper in the shoe works wonders. Rotating a few pair of socks also helps a lot.
I would not recommend buying Goretex shoes or water resistant shoes for your trip. They tend to be more false-hype then usefulness.
IMHO, you should get some waterproof shoes/boots for a multi-day trek on this specific trip. I'm inclined to give more weight to the Scottish respondent since I think their typical weather is likely similar to what you will experience in NZ. Lightweight, breathable shoes are great in warm weather with low humidity and little precipitation, but they are miserable if you don't have a chance to dry them out (i.e. if the air is cold and/or moist). I did an 8-day hike in southern Chile with my trail-runners and after the second day had developed blisters on both feet (this was with intermittent rain and several stream crossings each day, with moist air in the mid-50's), my buddy in waterproof boots was fine for the whole trip.
If it's only a comfort issue for you, and your trek is only a few days, then I'm not sure that it matters one way or the other. If this is a 9 day trek with lots of elevation/climate change, I think you should really consider getting waterproof boots, and maybe getting some conservatively styled models to maximize their utility (disco, anyone?)
It is a common expectation that, like in your daily routine, you can expect to keep your feet nice and dry for an entire multi-day hike. Gear manufacturers contribute to this perception by promoting equipment as waterproof, breathable, etc.
The fact is, if it rains, you can't. There are a number of reasons why:
Andrew Skurka says it here.
When it's wet outside, there's nothing I can do about keeping anything about me dry.
I have seen walkers (trampers in NZ) often ask rangers how to keep their feet dry in the rain. Every ranger I have seen answer that has responded with "Unfortunately, you can't".
But, there is good news.
Walking with wet feet is not so bad. Your feet warm any water and so you get kind of a squishy, wetsuit effect. I have walked whole days with wet shoes, which is fine as long as I have a plan to dry my feet at day end.
Walking the Kepler track (and most other trails in the region), there will almost certainly be a warm, dry hut at the end of the day. Plan on getting footwear that you can dry quickly i.e. a removable sole.
Take plenty of dry socks. Dry feet in nice clean dry socks at the end of a long rainy day are wonderful!
NB. I am presuming that you walk the Kepler Track in the summer open season, not the winter season. Winter season is a different story and not for inexperienced trampers.