How to send and reply to email
If you're new to professional emails, this is a good introduction. I've found that these conventions are often followed in the mathematical community.
The illustrated guide to a PhD
This helps to put the knowledge you gain in the process of getting a PhD in perspective. More generally, I really like Matt Might's blog–he has a really compelling story, and some interesting productivity tips.
Ravi Vakil is a senior mathematician at Stanford. Among other things, he is well-known for writing "The Rising Sea", an introductory algebraic geometry textbook. His advice for graduate students has been very influential for my path to and through graduate school.
Joel Spolsky, founder and longtime CEO of the company behind MathOverflow and Math StackExchange, explains one of the fundamental realities of software. The issue discussed here has come up in computational math research that I've done.
Things You Should Never Do, Part I
Another great Joel Spolsky article. Here, he writes from an industry perspective in the year 2000, but the main points still apply, and one can see them playing out in modern mathematical software projects.
The unreasonable effectiveness of the Julia programming language
Julia is one of my favorite technologies, and this article does a good job explaining why it works so well. The points brought up in this article, particularly the ability to easily mix and match with libraries on github (some of which are highly optimized), has come up multiple times in my own research.