Christchurch is the biggest city on New Zealand's South Island. It is not huge though and some part can easily be walked. You can even try hitchhiking while staying on a main axe with enough space for the cars to stop if you wish to get out of the city. In this case you should consider having a board!
From and to the airport
A bus shuttle to the city center is leaving from the front of the airport, but is really expensive. If you walk direction of the International Antarctic Center, you will find a bus stop that costs just a normal bus ticket price.
There is one big and straight road (Memorial Avenue/Fendalton Rd) that leads to the big park on the West of the city center. Is is pretty hitchable in both direction with a 'City Center' or 'Airport' board.
Walking can be a bit long (>5km).
If you're hitchhiking North you'd do best near the Motorway Entrance, but not too close to it, because the traffic entering the motorways moves quickly. There are traffic lights near the entrance, which aid in slowing down cars, and should allow drivers to get a better view of hitchhikers. Get yourself (by city bus or other means) to Belfast which is the suburb on the Main North Road immediately before the start of the motorway. Alternatively take a bus right out to Woodend on SH1 and hitch from the north end of town within the 50km/hour zone.
To get to Belfast take either bus 16 or the 'Northern Star' from the Bus Exchange.
On the map at Liftershalte.info <map lat="-43.514198796857" lng="172.6261138916" zoom="10" view="0" float="right" />
If you're heading South its best to do it from the southern end of Hornby just before the 50km/hr zone ends (just past the shopping malls).
From the center of town, grab the Free Shuttle (Red) until the last stop. From there grab bus 88 to Hornby. You'll get off next to a Robert Norris cafe. Walk to the main street, cross to the other side, and hitch south (HWY 1)!
Heading West you could try hitching from the beginning of Yaldhurst Road at Upper Riccarton or alternatively take the bus (or get dropped off) at the Yaldhurst Road/Russley Road intersection - this is at the start of State Highway 73 but there is a really good area for cars to pull over just past the intersection and before they get up to highway speed.
Other traveling infos
See the Nomadwiki article!