I've recently typed up a mathematics research paper, but I would like to make it appear more formal when I submit it to be peer-reviewed.
Also, how would I be able to incorporate LATEX into my paper?
Are there any templates out there that follow formats similar (maybe even exact) to examples such as this and this one.
You can download the latex sources of the vast majority of (actually almost all) math preprints posted to arXiv. Just go to the abstract page of your favorite math paper and click on "Other formats" under "Download." There you can find the link to the source file in latex as long as the author(s) uploaded it and complied by arXiv's latex engine (which pretty much everyone does).
I think reading the actual sources of nicely typeset papers you like is a very effective way to learn how to latex.
For example, here's the abstract page of my latest preprint:
http://arxiv.org/abs/1409.2559
and its source is here:
http://arxiv.org/format/1409.2559v4
To get the latex file I submitted, click "Download source" near the bottom right.
The file is in compressed format. But as the arXiv page says, your browser may uncompress the file. My Google Chrome for Mac does this, so what I actually get by clicking the download link is a latex file, which is named "1409.2559v4" with no extension. If your browser behaves the same way, you can simply rename the file to attach ".tex" at the end if the extension is important.
The question is based on something of a misunderstanding. On Stack Exchange, it is possible to incorporate LaTeX into a post by using dollar signs. However, when you're writing a stand-alone document, you don't "incorporate" LaTeX: rather, you write the whole document in LaTeX. (An analogy would be asking "How do I incorporate MS Word into a document?" You dont – you write the whole document that way.)
As such, you'll have to re-typeset the whole document using LaTeX, if that's the route you want to go down. The good news is that a LaTeX document looks a lot like a Stack Exchange post when you're editing it. It's mostly just a text file, with ordinary text as text and mathematics in dollar signs. (Plus a "preamble" of initialization commands at the top, and a few commands within the document, such as to start a new section.) So you can mostly copy-paste from your Word document (or whatever it is you used for the first version), put dollar signs around short pieces of maths that appear in the main text (e.g., Pythagoras proved that $a^2+b^2=c^2$.) You will have to re-work the displayed equations and any more complex pieces of inline mathematics.
I would recommend using the amsart LaTeX format: http://www.ctan.org/pkg/amsart
If you would like an intro to LaTeX itself, consider going through the wikibook: http://en.m.wikibooks.org/wiki/LaTeX
Often, the journal you want to submit to will provide a LaTeX template. Look at its "Guide for Authors" section.
