The central city is compact enough to walk for most sightseeing purposes. Cathedral Square, the Botanic Gardens, the Art Gallery, the 185 Chairs memorial, and the Cardboard Cathedral are all within reasonable walking distance of each other, though the city's spread means you will cover more ground than a map suggests. A good pair of shoes is useful; the rebuilt city has some rough surfaces in areas still mid-development.
Metro buses are the main public transport network and connect the centre to the suburbs. Route maps and timetables are available through the Metroinfo website and app. The Christchurch Tram runs a tourist loop through Cathedral Square and the city centre — it is primarily for visitors rather than getting anywhere quickly, but it is a pleasant way to see the central streets and the running commentary is genuinely informative. A day pass is reasonable value.
Cycling has improved significantly since 2011; the rebuild included dedicated bike lanes on many central streets and the riverside cycle paths are excellent. Bikes are available for hire from several operators near the Botanic Gardens and Cathedral Square. Uber operates throughout the city and is reliable. Taxis are available but pricing can vary.