Keepers and vendors are the most important - mechanics wise - keepers keep the animals from starving, vendors keep guests happy. Janitors also help on guest satisfaction (you can assign them to: bathrooms, bins (and walkways), and animal transport). Vets are necessary for animal research (getting more enrichment items) and mechanics are necessary to keep barriers fixed, and research new building themes. A thing that even experienced players sometimes forget is that when a new building theme is implemented in the game (like the new Oceian theme), you need a mechanic to research it if you're not in sandbox. Animals will never die (or even really react) to being ill, so whether you care about that depends on your preferences. Vets are the only staff members who can remove dead animals or capture escaped ones.
For me, the most 'fun' of the staff are the educators and they do provide a big education boost for guests. They have a variety of unique animations (speeches during animal talks, holding some exhibit animals during talks, the walking tour and - if you can catch it - occasional casual talks as they're roaming the zoo.
Remember, within the menu of each staff member there's a tab that allows you to select what type of activity each member does. I usually have a keeper that just cleans exhibits, for example. The poop scooper. You can direct janitors to just do bathrooms, or just move animals, etc.
Nowadays, I rarely bother with security. They really don't do much and if your guest happiness is high, there's little crime.
