Great questions. Whether or not you become a network administrator first depends on your general knowledge of networking and computer operations. If you already have a good handle on those activities, it may not be necessary to jump into network administration first. Then again, the job of "doing security" is mostly based on the job of "doing network and systems administration," so the latter is definitely a great preparation and background for the former.
That said, the job opportunities are best for Windows, but the tools and technologies for security are best on Linux or Unix, including Solaris, so if you really, really want to be a top-notch security practitioner you also need to be a Unix geek. Networking basics and user-level operating system knowledge is not even enough to scratch the surface yet, let alone plumb the depths of technology that a security professional must handle.
Hope this helps. Good luck with your career change and upcoming job decisions.
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This was first published in March 2002