Try to come out of the auto-mode and start handling the options found in manual mode like ISO, shutter speed, aperture, etc. Read the manual that came along with your camera, and refer to the guidelines to get a good hand on your primary equipment. Once you get to know how to navigate through the manual settings, you'll be able to customize every shot as per your liking.
If your camera comes with an external attachable lens, it is better to carry a decent zoom lens to maintain a safe distance between you and the subject. A handy waterproof backpack, along with comfortable shoe wear are a must-have for a quiet predatory walk near your subject. A compact lightweight tripod can help in getting the perfect steady shot.
If you want the subject to be still while keeping the background a little blurred, it is advised that you take the shot at high shutter speeds. The more challenging the shot is in terms of movement, the higher the shutter speed should be. Adjusting the proper shutter speed manually for capturing the perfect shot will take diligent practice.
Keeping the ISO value at an optimal level prior to taking the shot is quintessential. Ideally, the ISO value should be inversely proportional to the day-light availability. The less amount of natural light in the environment means you should set the ISO value at a higher level. You can use the Auto ISO setting, but it may sometimes result in noise factors. After taking a good amount of pics, you'll get a good idea as to what the ISO value should be set at.
Don't get caught up in the intricacies of photography, so much so that you miss the bigger picture. Whether you're on a wildlife safari or just for a trip around the zoo, take your time to enjoy the moment, relax a bit, and then opt for clicking your desired picture. "There are no rules for good photographs, there are only good photographs." - Ansel Adams