Companion
Give your animal a companion when it gets lonely, but not a mate. A companion of the same sex will cheer it up without producing babies for you to handle.
Avoid Crowding
Place zoo paths one square away from the animals' exhibits. This will stop the animals from complaining about being overcrowded by the zoo's visitors.
Shy Animals
On some parts of the animals' homes, construct fences that visitors can't see through. This will keep shy animals happy.
Unlock Foliage and Toys
Spend money on research programs such as "Animal Enrichment" and "Exhibit Foliage." Animals are at their happiest with their favorite toy or plant, but not all of these are available at the start of the game. The research programs will unlock these items.
Suitable Exhibits
Click on an animal to view its information panel. Look at the "Tree Bar." If it is yellow, your animal is not satisfied with its exhibit. Select the "Zookeeper Recommendations" button, and change the animal's home to whatever he advises.
Larger Exhibits
Build your exhibits larger than seems necessary. If the animal gets a mate and has babies later in the game, its home will be too small and it will become unhappy.
Foliage
Press "Ctrl" and "G" together to bring up the map grid. Use the "Foliage Tool" to completely fill tiles with your animal's favorite plants. It is important to use this method because some types of foliage only fill one quarter of a square when using the normal technique.
Herbivores
Acquire more herbivores for your zoo as they are much easier to keep happy than other animals.