A few weeks ago, I attended an IMA workshop on careers in mathematics and talked to graduate students and postdocs about this topic. The conversation focused on what actions to take as a graduate student that can be helpful in various jobs in the mathematical sciences. I summarize my thoughts here.
Familiarize yourself with what employers in industry, national laboratories and academia are looking for: This can be done by reading job ads in various places and summarize the requirements and preferred skills they list. In industry, many employers look for problem solvers with programming and data analysis experience. They often list desired qualifications as collaborative, independent, self-motivated, and good communicator. In fact, national labs list similar qualifications in addition to specific technical requirements. Willingness to learn is a must. I recommend developing the following dispositions:
Be curious in many areas of mathematics and pursue this curiosity. It helps to attend seminars and colloquia regularly and talk to the speakers afterwards. You can ask for references to interesting issues. It is a good practice to read papers and books on issues that are tangential to your main research topic in order to develop breadth. Learning significant theorems in various areas helps understand current issues in a historical context.
Make connections between different areas of mathematics. I recommend writing down ideas in a journal for easy reference later and making note of the connections you see between different areas of mathematics. Can results in one area be translated to another area? It is interesting to take detours from your main work to explore these connections.
Learn programming. MATLAB is ok but employers outside academia usually ask for Python, R, Java, C++. National labs use FORTRAN too.
Become proficient in statistics. You will be more desirable if you know how to use statistics to uncover correlations and patterns in data. Employers use these patterns to make business decisions.
Learn to write and speak. Good communication skills, oral and in writing, are extremely important in all jobs. Successful grant proposals, journal articles, professional reports, conference presentations, and product pitching require excellent communication. It is possible to improve these skills by reading and listening to others to determine what is effective and what to avoid. Consider different ways to present results and practice with a mentor who can give you feedback.
Take summer internship at National Labs or industrial companies while in graduate school. This is a good way to develop long-lasting collaboration with other researchers and to experience the work they do.
Talk to people at conferences, talk to visitors at your institution, expand your professional network. This is one way to understand the current limits of what is known and the key questions in mathematics. If you make good contacts, follow up on promising collaborations and aim to develop a body of research that combines collaborations and individual work.