I am an independent researcher (no academic affiliation).
If I submit a mathematical research paper to a journal or conference, how do I get any subsidies, grants, or funds for this work? Are you supposed to write a grant proposal?
What exactly would be the best course of action for an independent researcher to take if they would like to publish their paper and get paid a salary?
Publication in many reputable Mathematics journals is free to the author and paid for by subscription. There are now many reputable journals who use an open access model, where the authors pay for publication, but the article is free to readers. There are many more predatory journals out there who use the open access model, but do not do the traditional jobs of journals (of peer review and good editing). If you are worried about money, do not use the open access model.
If you are submitting to a conference, you promise (by the fact that you are submitting) that you will travel to the conference on your own, pay conference fees, and present your paper. While many attendees will pay for this out of a grant, a large number will still have to pay these costs out of their own pocket, as many departments do not have sufficient "travel" funds. Some conferences have found sponsors that allow them to support attendees, but this is usually limited to students.
In your situation, you need to first select a publication outlet. You can use google scholar to find papers that are similar in topic to yours and see where they are published. You will then follow the normal submission process, which is now usually electronic. You will be asked to specify an affiliation, so you will have to fill in your home address for that. The editor (or probably an assistant to the editor) will take a look at the paper. If your paper is not following the conventions of Mathematics, your paper will receive a "desk reject". Use papers with a similar topic as your style and content guidelines to avoid this. If the editor thinks that your paper MIGHT have substance and be worthy of publications, it will be send to a referee.
Please be aware that mathematicians at good universities and editors receive "crank papers" by usually well-intentioned people that do not understand Mathematics. For instance, there is a rather complicated result (that is nevertheless proven to most undergraduates in Mathematics if they take Algebra) that angle trisection is impossible. This has not prevented many people to think they solved trisection, usually by coming up with a good approximation. However, good approximations are known and are not interesting in the broader context of Mathematics. I am not saying that you are falling into this category, but if I as an editor received a paper authored by someone with no affiliation, I would be suspicious of the paper being a "crank paper". For you, this implies a need to be careful in how you write and state things. Be also aware that Mathematics is cumulative and that many things have been discovered and rediscovered by good mathematicians. If your contribution is good, it might still be well-known, for instance because it is a trivial application of a theorem. Knowing the literature in Mathematics is far from easy.
It is also a good idea to find someone with a Mathematics education to read through your paper first. Finding a Mathematics professor to take on this task will be very difficult because this is hard work without reward for them.
However, funds are hard to come by and even more for someone without affiliation. Agencies like the NFS in the US would have a hard time to give money to a private person as would the private person complying with the grant conditions. In Europe, grant administration tends to be even more complicated. Without a published record, it is not worth while asking for money. But these grant would be to support future work, not publication of already obtained funds.
TLTR: Submit to a decent journal in Mathematics carefully selected. This will not cost you money. Save time and effort by getting feed-back on your manuscript first.
As the answer of tschwarz points out, many reputable publishers will publish your work without cost if you are an independent researcher and don't have funds to cover the normal fees. You don't earn anything from submission to journals or conferences. You aren't "selling" your work. What you "earn" is reputation.
But the question of earning a salary for work in mathematics is quite a different issue.
For that, you probably need to get one or more degrees in mathematics and get hired by a university or a research arm of some major corporation. The most common path is with a PhD and a faculty position. The road is long and arduous. This path is open (but crowded) for both pure and applied math, though probably biased toward pure math. If you don't have any degree yet, the path takes about ten to fifteen years to traverse, but it can be interesting along the way - and sometimes frustrating.
Some people can get by with only a masters or even a bachelors degree if they have skills suitable for some corporation. But those jobs are much fewer. Some pay very well and others not so much. Most corporate research is in product development, not pure research. Applied math is more suited to this.
At the outside, if you have done something that can lead to a patentable idea, it might be possible to form a company on your own to exploit it. Or even sell the patent to an existing company. This is even more rare. Again, applied math might put you on this path.
