This is a great question, with many possible answers. Here is what I try to achieve with my writing and are some of the guidelines that I try (and usually often fail) to follow in my writing:
- With mathematical writing, as with any writing, think clearly about the story you are trying to tell. Consider why this might be interesting and ruthlessly remove even the most fascinating parts that do not push the story forward.
- Watch your language, watch for any words that have a use or meaning in mathematical writing, there are some surprises ("admits" for example) try to remove these, or justify why you need them.
- Do not be frightened to go slow, mathematical ideas we understand well can often seem trivial. Yet also respect your audience, tackle the big ideas. Remember that you are communicating the general ideas and not the specific technical details.
- Be concrete, an example can often show an idea off well, and do not forget counterexamples, these can often be more informative than the ones that fit it.
- Finally, whenever possible, use a picture, they can allow people far deeper into the mathematical ideas than words alone.
Good luck with your wonderful experiment.