Direction

There was a time, not so long ago, when most zoo directors came up through the ranks within the institution. These were often biologists or veterinarians who held various roles until they eventually rose to director after many years of service and leadership. To be sure, many current (and probably future) zoo directors took this route. However the times are changing.
Managing a zoo is an increasingly complex undertaking. Beyond overseeing a staff of hundreds, zoo directors are also tasked with leading international conservation efforts, overseeing enormous budgets and varied revenue streams (i.e. tickets, concessions, membership, gifts, events, municipal contributions, etc.) and launching multimillion dollar fundraising campaigns to build modern exhibits. They also have to manage a diverse group of stakeholders, including members, local government, conservationists, activists, donors (both corporate and individual) and their own employees. Just because you love animals, doesn't mean you'll love multiple weekly finance meetings about your zoo's operating budget :) Consequently many zoos and zoo boards are now looking for leaders with substantial general management experience in addition to a clear passion for and commitment to animal welfare and conservation.
Long story short, working your way up through a zoo is a fine option but so is building your leadership skills in a corporate or large non-profit environment. Just make sure you continue to learn about and contribute to the zoo world while you develop professionally elsewhere.
Finally if you truly want to be a zoo director, and this goes for many other zoo professions as well, be prepared to move wherever there is an opening. There are about 230 zoos and aquariums accredited by the Association of Zoos and Aquariums (AZA) in North America and most of these are not in major cities (think the opposite). Zoo directors often stay in their roles for many, many years so openings are limited. Many zoo directors in major cities were offered the role after doing an outstanding job at a smaller zoo somewhere relatively remote.