Assume an undergrad has something worth publishing: how might (s)he get it published?
Here's what you should do:


Find an appropriate journal with a double blind review process.
Submit your paper.
See what happens.


Double blind means that you do not know the identity of the reviewers, and they do not know your identity.  The ostensible purpose of this setup is to encourage reviewers to evaluate the paper based on its academic merit, rather than the pedigree of the author(s).

In reality, of course, there will always be politics in reviews, and many reviewers will nonetheless formulate theories about who you are and where you're from.  Moreover, things like hot trends in the field can influence judgement—which is why it's important to identify a journal compatible in spirit with your chosen topic.

You will also be judged on the style of your writing, and the visual style of your submission.  Reviewers will be much more comfortable accepting a submission that looks like a published article than one with strange typography, layout, etc.  E.g., don't submit a paper written in MS Word when everybody else in your field uses LaTeX.  Strange as it may seem, you will look like a nut job!  Even different linguistic tone and structure can make your reviewers think you are a quack, even if your idea is legitimately awesome.  In short: be a conformist when it comes to your first few submissions.  Once you understand how the game is played, you can start to break the mold (hopefully for the better!).

Even accounting for everything above, there is an extremely good chance your submission will get rejected.  Coming up with genuinely good ideas is hard; developing political/cultural/intellectual savvy in academia takes time and experience—especially as a lone undergrad.  And rejection will sting!  But the submission process will be a valuable experience nonetheless.  After a few days, go back and read your reviews calmly, and think carefully about why the reviewers wrote what they did.  Was the typography too bizarre?  Are they protecting their political interests?  Or is your idea simply not as good as you thought it was?  The more papers you submit, the better you will get at answering these questions objectively and honestly.  Eventually, you will master the game and can just focus on the work.  I hope.

Good luck!