It's important to have a degree related to animals. The more education you have will provide you with more opportunities for advancement and more pay. You should also continue your professional growth by joining enrichment committees. Here are a few: AAZK (American Association of Zoo Keepers), AZA (Association of Zoos and Aquariums), EMA (Elephant Managers Association), ABMA (Animal Behavior Management Alliance).
Also, you need to be CPR and First Aid certified. You never know when you might have to give mouth-to-mouth to an aardvark.