The biggest tourist attraction in Oamaru is the little blue penguin colony at the end of Friendly Bay, but you find little penguins all around New Zealand. NZ has three types of penguin. The yellow-eyed penguin/hoiho (endangered, with numbers crashing, possibly due to fishing activities), the Fiordland crested penguin/tawaki and this little penguin/kororā.
I used to volunteer as a penguin guide at the breakwater in St Kilda when I lived in Melbourne, so I saw plenty of little penguins there too. I still have a scar on my lip as a souvenir of an up close and personal encounter (when I was holding a penguin for the penguin researchers to measure). So I wasn't prepared to pay to go and see the penguins in Oamaru, but since I knew the type of place penguins like to nest, I walked to the beach in the evening and found plenty wandering around.