As a newbie, most of what you need to know will be explained after you get hired. Unlike long-shoremen or plumbers there is no secret handshake requirement to get a job. Unlike other professions, the airlines hire pilots, not the unions. There are really only a few job-search issues related to unions which you need to be aware of...

1) There are several unions out there... some, called in-house unions, are unique to only one airline (SWA, AA). There are also two large national unions: ALPA and Teamsters. Which union you end up with is almost not worth worrying about (although I personally have a few issues with the teamsters).

2) Non-union shops: There are several non-union airlines out there...lack of union can be an indicator of sub-standard work rules and pay, but you would really have to look at the specifics of the company in question. I wouldn't absolutely rule out a non-union shop, but be careful.

3) The one thing that MIGHT trip you up in your job search could be at an interview if the topic of unions comes up (I'm not even sure they are ALLOWED to discuss this). If you're talking to an interview captain you might have to walk a fine line...you don't know whether he's a big union boy, or a management captain who dislikes the union. I'd avoid expressing too strong an opinion either way unless you know where the guy's coming from. Since you're coming from the military, nobody's likely to expect you to have much knowledge or opinion on unions anyway.