Best options for publishing original research without being in academia? What are the best options for publishing an original piece of research for someone who is not in academia? The main issues are:


zero prior publications, i.e. no reputation whatsoever in the given field,
no contacts, i.e. no-one that can step in as a "referee" or a "supervisor",
no academic titles in the given field (ex. a PhD or something equivalent).


The research is original and does provide new information about a given topic. But given the above limitations the options are not obvious.

What can one do in this case? I heard people who are starting out in publishing can post on arXiv.org, and once they have a couple of "published" papers respectable journals will be more likely to consider the candidate.

Is there any advice out there for people doing research as hobby?
My advice: Work on the second point, get contacts.
Send an email to people working on related topics, ask them if they would like to take a look at your work, if you can ask them a question, etc.
If possible, try to contact PhD candidates or recent post docs, as they often have more time available than a professor and might be more willing to look at what you did. For that you should have a nicely written, complete draft available, that is both short and informative and that is nice to read - for example: don't put way to long equations, try to drop these and state that "one can prove, that...", maybe adding that you will send the proof if needed.

Once you have established a contact, the first thing should be to find out if what you did is really worth a publication. Maybe it is already known and you simply don't have access to the journal? Maybe it is not interesting enough, e.g. it might be considered a simple homework exercise for someone who is an expert on the field?

If the expert agrees that what you did is indeed new and a nice result worthy of being published, ask him for advice to proceed. If possible, get him as a co-author or ask him to refer you to one. Such a co-author might take care of things like


Literature review.
Explaining why the found results are important and have an impact on topic X.
Which journal to send to?
Given his (known) name, journals will no longer reject due to unknown author.
...


Of course you would not be the single author anymore and people might think that your co-author came up with the idea. How to deal with this topic depends on what you plan to do next: Do you want to go for an academic career? Do you say you don't care as long as people get to read your results, because you are only doing research in your free time and it will most likely be years until your next publication?