Integrity without knowledge is weak and useless, and knowledge without integrity is dangerous and dreadful. (Samuel Johnson, “Rasselas”)
Take your duties and responsibilities seriously; being frivolous is fine with friends, but can be annoying for your colleagues, especially those who are busy with similar responsibilities.
One’s writing should also be taken seriously; your work is going to appear in permanently available journals, and what may seem witty or clever today may be incredibly embarrassing for you a decade from now.
Being assertive is fine, but being overly self-promoting or competitive is generally counterproductive; if your work is good, it should speak for itself, and it is better to spend your energies on creating new mathematics than trying to fight over your old mathematics.
Try not to take any research setbacks (such as a rejection of a paper, or discovery of an error) personally; there are usually constructive resolutions to these issues that will ensure that you become a better mathematician and avoid these problems in the future.
Be generous with assigning credit, acknowledgements and precedence in your own writing (but make sure it is assigned correctly!). The tone of the writing should be neutral and professional; personal opinions (e.g. as to the importance of a subject, a paper, or an author) should be rarely voiced, and clearly marked as opinion when they are. In short, you should write professionally. (See also my advice on writing papers.)
On your web page, keep the personal separated from the professional; your colleagues are visiting your web page to get your papers, preprints, contact info, and curriculum vitae, and are probably not interested in your hobbies or opinions. (Conversely, your friends are probably not interested in your research papers.)