This isn't like getting a cat or dog; you actually need to have some education and training before anyone will let you near one of these very valuable, possibly dangerous living exhibits. It's important to have a degree related to animals. Zoology and biology are just a couple, but there are certifications you can get (especially at the graduate level) that'll allow you to focus on specific animals with specific needs (source).
It goes without saying that a strong educational base will provide you with more opportunities for advancement and increases in pay, but that's not all. You can also continue your growth by joining enrichment committees like the American Association of Zoo Keepers, the Association of Zoos and Aquariums, or the Animal Behavior Management Alliance.
Also, you'll need to be CPR and First Aid certified (source). We're not kidding—you never know when you might have to give mouth-to-mouth to an aardvark.
Okay, we lied and actually were kidding. The CPR is for the people who visit the zoo. CPR on an aardvark, can you imagine?