Canterbury Museum sits at the entrance to the Botanic Gardens and has been collecting since 1867. The building is a handsome Victorian Gothic pile that survived the earthquake in reasonable shape, which is fortunate because the collections inside are substantial. The Antarctic galleries are the highlight: Christchurch was the departure point for most of the major polar expeditions of the heroic era, and the museum holds artefacts, equipment, and photographs from Scott, Shackleton, and Amundsen's Antarctic journeys that you will not find anywhere else with quite this density or context. Scott's sleeping bag is here. So are his boots.
Also worth tracking down is Fred and Myrtle's Paua Shell House — an entire room from a Bluff home whose owners covered every surface, including the furniture, in iridescent paua (abalone) shell over several decades. It sounds overwhelming and it is, in the best possible way. Admission to the museum is free, which makes it easy to take your time. Allow at least two hours if you are interested in either Antarctica or New Zealand natural history.