This may not sound like what you're looking for, but it's how I got started.
I'm going to assume (for purposes of this reply) that you're already within some sort of IT job function. Otherwise, you'll have to get their first.
First off, view everything you touch job-wise as it relates to security. If you work for a company that has security people, ask them if there are any documents, policies, guidelines, etc. that can help you, as an individual contributor to better follow whatever standards are in place. Just don't gush.
When you see something that looks "odd" from your "security" point of view, check it against the existing guidelines. Or ask security people (in your enterprise) what they think of it. Don't over do it - lest you be viewed as NOT doing YOUR job. This is not going to happen overnight or even over-month (to coin a phrase). But the more you focus on security aspects, while still getting your normal job done, the more you will come to be regarded as part of the security effort.
One Caveat - some "security" people might view you as a threat or upstart. If you've got some of those in your company, you'll have to tread more carefully. This might especially be the case where they know that either they're not doing the job right, or they're frustrated by the lack of support they get from upper management. I've seen some of these scenarios, but don't have an easy answer for that.
Read - lots. If you don't already, subscribe to the SANS.org emailing lists, Computerworld security email list, and many others that are around. If you wind up being the person who has alerted your outfit to new threats - and particularly - how to deal with them, you become more valuable in a security light.
Spend your free time working on related things - like learning snort (open source intrusion detection system). Prepare to be frustrated, and learn patience and persistence.
I've just barely touched the surface. Partly because there's so much, partly because I've answered queries like this before, and it gets tiring doing it multiple times.
Search the archives of this server - you'll probably find some relevant material there.