Christchurch Airport: The useful public-transport option is an InterCity coach to Oamaru (check whether your chosen departure collects at the airport; otherwise take the airport bus into central Christchurch and board there). Allow roughly 4½–5 hours airport-to-Oamaru and about NZ$30–50 for the coach. A rental car is better if Oamaru is the start of a Waitaki Valley road trip: SH 1 is about 250 km and around 3 hr 15 min.
Dunedin Airport: There is no fast, obvious airport-to-Oamaru public-transport transfer. For a one-stop itinerary, rent a car and allow about 2¼ hours door to door; Oamaru itself is about 115 km/1½ hours north of central Dunedin on SH 1, but the airport is south of the city. Do not take an airport taxi all the way unless several people are splitting a pre-quoted fare—it is an expensive use of a taxi.
Oamaru bus stop, Eden Street: This is the practical arrival point for intercity coaches, just east of Thames Street/SH 1. It is walkable to the central accommodation, Victorian Precinct and harbour. Daily coach services connect Christchurch and Dunedin, and Christchurch–Te Anau services also call here. Although the Main South Line runs through town, do not plan around it as a regular passenger-rail arrival.