This advice is mainly for first-time authors.
A journal article is going to be a permanent record of your work; thus it is important that these articles are as professionally written as possible, so as not to cause embarrassment several years from now.
Thus the majority of the paper should be objective and factually based; informal remarks and opinions are definitely permissible (and encouraged), but should be clearly labeled as such to distinguish them from formal, rigorous assertions. (For instance, they can be placed in footnotes or in a remarks section.)
Overly philosophical, witty, obscure or otherwise “clever” comments should generally be avoided; they may not seem so clever to you ten years from now, and can sometimes irritate the very readers you want to communicate your result to.
Mathematical papers need to have a properly formatted title, abstract, introduction, and bibliography. (Note that the format of a mathematical paper is slightly different from a scientific one; for instance, one usually does not have a “Conclusions” section, although a “Further remarks” or “Open questions” section is not uncommon.) The references should be current, showing all recent related work; even if these works are not directly used in your paper, a comparison between your approach and others in the literature is expected. These references should be cited within the text whenever appropriate, giving an accurate assignment of credit, provenance, and precedence.
The standard format for mathematical papers is TeX, AMS-TeX, LaTeX, or AMS-LaTeX; other formats such as Word or Mathematica can cause technical difficulties (and will ultimately need to be converted to a TeX format), and so should be avoided. If you intend to go into research mathematics as a long-term career, it is definitely worth investing some time in gaining a reasonable level of competency in some form of TeX (I personally recommend LaTeX).
Spelling and grammar should be checked, especially if the language used (most probably English) is not one’s native language; in such cases, one might consider using spell-checking software. The English language, when used correctly and thoughtfully, can convey some subtle nuances and modulations which can greatly assist a reader in grasping the key points and insights of a paper; conversely, a careless or incorrect use of English can lead to confusion or misunderstanding on the reader’s part, and conveys the impression that the paper itself is also careless and incorrect. So it is worth devoting some effort and thought to ensuring that the prose portion of the paper is at a professional standard of quality, though as mentioned before one shouldn’t try to be excessively polished or clever in one’s choice of words; we are, after all, taking about a mathematical paper here, rather than an essay or a piece of literature.
In order to be suitable for a research journal in pure mathematics, any argument in one’s paper which is central to one’s main results must be backed up with rigorous proof (or rigorous numerics), unless it is an argument already in the literature (or otherwise very standard), or a very slight variation of such an argument (in which case one should cite the literature and describe the necessary modifications to the argument). Note that one is certainly allowed to use the English language while writing rigorously! One can also certainly supplement one’s rigorous arguments with more non-rigorous discussion, motivation, or heuristics.