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First, said average joe must know the field, since it's otherwise difficult to know that something is a breakthrough - it could have been considered and rejected some decades previous.

Second, use a site like Google Scholar to find similar work in the area, or go to the library and do the same. This gives average joe an idea of which journals are relevant to the field.

Third, pick an appropriate journal, go to its web site, and follow its instructions to the authors when writing the paper. It's a good idea, though not required, to read some of the recent publications in that journal to get a sense of its style.

Another concern here is that some papers require a publication fee, and others don't. Some of the ones which require a fee are also disreputable, and will publish essentially anything in order to get the fee. See http://www.biosciencewriters.com/Selecting-a-Target-Journal-... for more details about how to select a journal.

Fourth, send in the paper, or as that previous link suggests, send in the abstract and ask the editor if the journal is interested in a paper of that sort. The paper (or abstract) submission might be rejected by the editor or reviewers, in which case it's a matter of picking another journal or improving the paper based on the comments. Or it might be accepted with some changes needed, or it might be accepted outright. In which the paper is published.

There are many resources on "how to write a scientific paper", such as http://www.phys.unsw.edu.au/~jw/paper.html and even a book with the title "How to Write and Publish a Scientific Paper", which you can get from your library either directly or through InterLibrary Loan.

To increase the chances of recognition, average joe could also do a good job of making the paper readable and reproducible, with details that are cognizant of the history of the field and the reasons why the new discovery should be considered "breakthrough" rather than "novel." Said joe could also attend relevant meetings and talk with others who are interested in the topic, and even present that information.