Being a researcher requires developing a number of skills that are not part of the standard curriculum of academic courses. This includes:
Writing scientific papers
Preparing posters
Delivering oral presentations
Reviewing papers
Sharpening these skills can make a significant impact on the exposure of your work and help build reputation.
Over the last few years I have come across valuable resources that provide concrete suggestions and tips from highly respected scholars, which I am happy to share:
Writing papers with mathematical content by John N. Tsitsiklis contains much needed guidance on effective writing for young researchers (and a few more helpful pointers appear in the link)
Delivering oral presentations by Patrick Henry Winston is a MUST-watch video!
Mathematical English by Jan Nekovář is a concise introduction, filled with examples, for describing many common mathematical terms in English
Writing papers and giving talks by Wheeler Ruml is a helpful video containing various high-level suggestions
Advices for PhD students by Sven Koenig is an hour-video filled with gems
Oral and poster presentation by Reuven Boxman gives many practical tips for preparing and delivering presentations and posters, with a focus on getting the right attention at conferences
Mistakes reviewers make by Niklas Elmqvist aids in understanding the peer review process, and provides concrete guidance on how to be a better reviewer
A few more practical tips of my own:
If you have the option to participate in a doctoral consortium that is held under the umbrella of a conference that relates to your research field, then you should definitely go for it. These are events that have curated materials for PhD students, with the goals of increasing exposure to the relevant research community (which is important!) and providing with concrete tools for doing more effective research. Direct feedback from experienced scholars in your research community is valuable! From a timing perspective, it is more useful to aim at participation after obtaining some results which you can share, which is typically a year into the PhD.
Concentrated summer/winter schools in topics that are of interest to you are also useful, especially as they usually don't require preparation, which makes them low-overhead and high-gain. A fine combination.
Teach a course! Even if you are not thrilled about the idea, it is still worth holding a teaching position in a university for at least a semester or two. I guarantee that you will learn a lot from the process. Research and teaching are closely related, so there is significant cross-fertilization.