Ask zookeeper ANIMALS - he'll mention something about procuring animals for the zoo. Go to the outback and GET EUCALYPTUS TREE when you see a eucalyptus tree. Find a koala and GIVE EUCALYPTUS KOALA, and it will follow you. Take it back to the zookeeper for xp.
Go to the Lucky Nugget and buy a loaf of bread. Go back to the outback and GIVE BREAD WALLABY. It will follow you. Take it back to the zookeeper for xp.
Go back out to the outback, and find a kookaburra. Follow it until it's in a room with a tiger snake. KILL SNAKE. The kookaburra will fly down, eat the snake, and then follow you. Take it back to the zookeeper for xp.
I have a vague recollection from doing this years ago that sometimes it takes more than one leaf/loaf/snek to get the animals to follow you, but when I tested it the other day, they followed me on the first go.