Christchurch International Airport is 12 kilometres north-west of the city centre and is the main gateway to the South Island. Air New Zealand and Jetstar operate frequent domestic flights from Auckland (around one hour) and Wellington (around one hour), with multiple departures daily. International routes connect directly to Sydney and Melbourne (around three hours), Singapore (around eleven hours), and Doha (around seventeen hours, with Qatar Airways).
From the airport, the Purple Line Metro bus (route 29) runs into the city centre and takes around thirty minutes. Shuttle buses run to central hotels and are bookable at the airport. Taxis and Uber are readily available. If you are driving, the motorway into town is straightforward, and rental car companies are well represented at the terminal.
By train, the Northern Explorer from Wellington terminates in Picton; from there the Interislander or Bluebridge ferry crosses Cook Strait (roughly three hours) and you can pick up the Coastal Pacific train to Christchurch, a scenic route down the east coast of the South Island. It is a long journey but worth considering if time allows.