Holly, I try to stay as closs as feasible to eye level with the animal and the get at a 3/4 view or a side view. Head on shots work sometimes but the 45-90 degree angle range seems to give the best shots more often.
The next thing you need is a little patience. Many animals are put off by cameras because of tyros throwing flash in their face. I try to never use flash. Also if the sun is at your back you eliminate the sun glinting off the front element and driving the animal away from you.
Also if it's crowded and you see someone who the animals are avoiding position yourself as far from them as you can andd the animal will likely come right towards you. That's how I took most of the silverback gorilla shots.
Animals are very intuitive, especially the higher simian species, and seem to have an inate ability to sense who is a putz and who is there to peacefully coexist with the animals.
Another tip is if you can find a zookeeper that knows the feeding schedule. If you arrive right after feeding, much like humans, they tend to be inactive. OTOH they have an internal clock that knows when feeding time is near and if you are just ahead of lunch they are quite active.
If there aren't a lot of people around you will find that the same types of sounds that get a baby's attention SHHHHHH and cooing sounds work well with animals as well. Once they don't fear you it's all easy. Too many people however terrorize the animals with too much flash and camera lenses into the sun which can be blinding. I don't know, but suspect, that many animals who were captured in the wild relate a telephoto lens to a weapon at first.