What would you advise someone who wants to do research as a hobby?
If you intend to publish it is essential that you learn how to write academic papers. Even if you can already write well, it still takes effort to get the style, focus, and, in particular, the quality right. Academic writing is highly compressed, rather formal, and precise. Theorems and so forth must be written in a particular way. Writing proofs is an art. There are even (implicit) standards for referencing the literature. Get any of these wrong and your article looks amateurish, which lends itself to rejection irrespective of the quality of the contents. 

Here are some general tips that will help in this direction:


Use LaTeX. Papers, especially formal papers, written with Word look like crap.
Read a book on academic writing (and do the exercises). We use Academic Writing: A Handbook for International Students, mostly because our students are non-native English speakers.
Read Writing for Computer Science: The Art of Effective Communication  by Justin Zobel.
Learn how to write theorems and proofs correctly. One way of doing this is to find a high quality book in the area and mimic its style, even if it comes down to copying the text word for word to get the feel for how things are written. Then when you write your own results, consult the book regularly so that you can mimic the style. I know people who have used Relation Algebras by Games by Hisrch and Hodkinson as an exemplary model of mathematical writing. Undoubtedly there is such a text in your area.
Learn the conventions of the venues/communities where you plan to publish. Most papers have an abstract and introduction, body, discussion, related work, conclusions and future work sections, but different venues/communities may vary the ordering or have different expectations about how much attention goes into which part of the document. Issues such as how much background information is added to your paper depend very much on the expected audience, and it always pays to know the background of the average reader in the community. Critically reading many papers from the target community and trying to comprehend the stylistic expectations, is the only way to address this issue (without a supervisor in the area).