Christchurch Art Gallery is the public contemporary museum/exhibition space. Look up Andrew Drummond, a NZ sculptor who works with coal, spagnum moss, brass, glass tubes, mysterious motors! His book is JUST published and probably worth hunting down.
The Arts Center, formerly Canterbury College, is a great area full of artists, restaurants and galleries. Most of it has reopened since the earthquake, but most of the artists studios are still closed. One of the amazing artists I met was Marilyn Rea-Menzies a tapestry weaver. This woman is amazing. I strongly suggest you go to her blog both for the art and the progress on clean up after the quake.