Coffee shops - certainly one of the highest per capita provisions around!
There are a number of museums, the national museum of Te Papa, the Wellington Museum of City and Sea as well as the WWI exhibition at the Dominion Museum.
There are a couple of large theatre venues, such as the St James and Opera house, with a number of small venues, The Gryphon, Bats and Circa.
Wellington also hosts or participates in the NZ International Film Festival, World of Wearable arts (WoW), NZ Arts Festival (which this year we had the Royal Edinburgh Tattoo perform at Westpac Stadium, NZ International Comedy Festival, a regular number of film festivals (British, French, German, Japanese, Brazilan etc) - usually organised in association with each nations culutral body, ie British Council, Allliance Francais, Goethe Institut etc.
A useful resource to find out what's happening can be found through the WellingtonNZ website http://www.wellingtonnz.com/discover/things-to-do/sights-activities/
An overview of cultural events
And an overview of festivals