If you want to capture a wild animals normal behavior then keep your distance and get a better lens, so they aren't reacting to your presence.
Adjust the angle of your shot to create the image you want. Lower your tripod and get down low to the ground to capture an intimate photo at the animals eye level. This will also make you less threatening to the animal.
While shooting wildlife it is best to shoot, shoot and shoot. Worry about the pictures later capture as many shots as you can while you can and keep your camera up so you don't miss anything. You can edit and view your photos when you get home. Don't get caught looking at your photo cache when a great opportunity is presenting itself in front of you.
Becoming well acquainted with your camera and knowing how to focus on a moving subject without a second thought will create the images you want. You don't want to be fumbling around with your focus while a bird is flying over your head. As the saying goes, practice makes perfect. Practice taking photos of animals that are less interesting to you, so you don't blow it the moment you are in the field trying to capture a more fascinating subject.
Patience is important for getting a sharp image of a moving animal. Having an animal in focus before they move is effective for capturing images of yawning or stretching. For a running or flying animal you want to pan along with their movement while shooting and follow through to keep the animal where you want within the frame. You can also anticipate the movement of the animal and have your camera set in focus and wait for the animal to move into the frame to capture your image.
Finding a good stage or background for your subject can make or break a photo. If you anticipate an animals movement (correctly) it gives you the ability to choose what the background will be. By focusing on an area that the animal will pass through you can set up the frame and wait for the subject to move into place then you will capture the image on the stage you want.
Knowing your subjects behavior is also important. Most animal activities are repetitive. If you know what to look for you can capture the action as it happens.