Unless you're responsible for a lion getting loose in suburban Ohio, you're not going to be making headlines any time soon—and if that's what it takes, then you probably wouldn't want to.
You might get a little bit of local fame with zoo-going regulars who watch your snake show every forty minutes. Another kind of fame will come to you in your circle of zookeepers, once you speak at a few conferences. (And yes, those conferences exist.)
Regardless, you won't have to worry about dodging the paparazzi any time soon.
If you've got the right personality and you work at the right zoo, you might even get some calls to visit talk shows with a van full of critters. You'll love it when they start calling you the next Jack Hanna.