As a network administrator, you have a strong start down the security career path. It's essential that you have a strong technical background to gain "street cred" in the security profession. My first recommendation would be to get as much exposure to the concepts and technologies of network security as you possibly can. Such experience should be a natural extension of your existing position, so the task shouldn't be too hard to accomplish....
Learn about firewalls, intrusion prevention systems, network authentication and any other technologies you can get your hands on.
You don't mention anything about your educational background in your question. If you don't already have a bachelor's degree, take the time to earn one. I don't know many people in our field (although there are some) who don't have at least an undergraduate degree.
Finally, you'll need to get a broad exposure to information security practices and principles. There are many ways to gain this exposure, either through classroom training, self-study or hands-on job experience. Do whatever's available to you. The Certified Information Systems Security Professional (CISSP) curriculum is a great way to get a well-rounded exposure to our field.
Good luck in your career development!
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