As you have seen, animals come in many sizes and shapes. Animating them does not need to be a complex task. Try to understand the way an animal moves, through research and reference. Once you have the basics, filter those motions through the personality of the animal to turn it into a character.
Chapter One. Basics of Character Design
Chapter Two. Modeling Characters
Chapter Three. Rigging Characters
Chapter Four. Basics of Animation
Chapter Five. Creating Strong Poses
Chapter Six. Walking and Locomotion
Chapter Seven. Facial and Dialogue Animation
Chapter Eight. Animal Motion
Chapter Nine. Acting
Chapter Ten. Directing and Filmmaking