Where can I find templates for a mathematics research paper? 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.