Here’s some tips for visiting Wellington. It is true that living in Wellington one doesn’t necessarily see all the tourist things (true of any place), but we did spend some time touristing when we first came. (Hence many pictures here are from 2020…)
In central Wellington, everything is within walking distance. Wētā Workshop and the Skyline track are a bit farther out and would benefit from transit, though a few hours’ walk would get you there.
Transport landmarks and viewpoints
The Wellington Cable Car is pretty famous and provides good views of central Wellington and the harbour. It is next to the Botanic Garden. The Botanic Garden does the usual botanic garden stuff, including natives and imported plants.
Other viewpoints on Wellington are from Mount Victoria (easily accessible by car and bus) and Mount Kaukau (involves a decent hike up, elevation 400m). I actually quite like the Mount Kaukau walk (pics), but it’s not entirely trivial at 4km roundtrip with 300m elevation gain.
The Wellington Railway Station is another transport landmark, but unlikely to be useful as a tourist unless you want to get to the Kāpiti Coast or Porirua (or Auckland via the Northern Explorer tourist train). You could use it to get to the Wairapapa, but destinations from there are trickier (though there are wineries in Masterton).
I also highly recommend Zealandia. It has a lot of native birds, including the kākā parrot, and many other endemic (and some imported) birds.
For more nature, there’s also islands: Matiu/Somes Island, which is doable on a day trip, and Kāpiti Island which is a long day from Wellington (or an overnight on the Kāpiti Coast). There is also a pretty expensive overnight Kāpiti Island stay which I haven’t done.
OK, so we’re not frequent museum visitors, but Te Papa (or the Museum of New Zealand Te Papa Rongarewa) is the museum to visit in Wellington, or maybe even New Zealand in general. The permanent collection is free to visit; there are rotating exhibitions which are non-free. The permanent collection includes a colossal squid and other nature exhibits as well as exhibitions on social history (e.g. mana whenua the world of Māori; Pacific peoples; and about the Treaty of Waitangi) and Gallipolli.
Whenever I can, I go to farmers’ markets. The Harbourside Market is the only easily-accessible market in central Wellington these days (open Sundays till 1 in winter, 2 in summer). There was another market when we first came, but it got built over by condos. (It had more fruits and vegetables but fewer food trucks.)
We always get food at the Orange Chinese Food Truck there, as well as a rotating thing from elsewhere, and I get various fruits and vegetables. I have to admit that Kitchener Market is better.
Cuba Street is lively and contains the bucket fountain. Courtenay Place has a lot of restaurants but is a bit sketchy
There are various Lord of the Rings filming sites in Wellington and everywhere in New Zealand (there is the Lord of the Rings Location Guidebook by Ian Brodie). The Hobbit’s Hideaway is on Mount Victoria though it was highly CGI’d in the movie.
There are two “WELLINGTON” signs (like the Hollywood or Amsterdam signs). One of them is near the airport and makes fun of the wind, also near Wētā. The other one is near the Harbourside market.