Hi, keeper2be --
Generally curators and assistant curators work their way up into those positions after many years (a decade or more) of work as keepers, then lead keepers, then collection managers, and finally assistant curators.
As you've probably already found out, keeping is very competitive. There are often hundreds of applicants for any given position at the better zoos, and it takes a sharp resume to catch HR's eye.
Your education will definitely help you a lot! That's the good news. The bad news is that experience counts heavily in this industry. Volunteer experience, especially volunteering directly with exotic animals, will be a boost. But what most zoos look for in potential candidates is paid hours at AZA-accredited facilities. For entry-level people, you can often find seasonal keeper's-assistant type jobs posted on web sites like aza.org and aazk.org. These seasonal jobs are usually for the busiest months of the year (May through September at most places) and often involve work such as interpreting exhibits for the public, diet prep, some enclosure cleaning, and locking up after the keepers have gone home (summer hours are usually extended at most zoos). This is probably the best way to get a foot in the door in this industry.
You can also search for internships. There are some paid internships out there, which will rack up your AZA paid hours for your resume...but even unpaid internships are an excellent experience and a good thing to have on your resume. One of the greatest values in doing an internship is meeting people who you can impress with your hard work and knowledge. In this industry, most people know each other, even in distantly separated zoos, and word-of-mouth and referrals are your best friends.
Focus on building paid experience and on developing networking relationships wherever you can. Eventually you can work your way up to a curator position.