There are plenty of programs, often in the summer, to guide students through the process of doing original research in math. Career panels and other events of interest also take place from time to time. Below are some we’ve found; you can find others by Googling. In addition, several programs provide mentoring or other support for advanced studies in mathematics.
REUs - https://www.nsf.gov/crssprgm/reu/list_result.jsp?unitid=5044, https://sites.google.com/view/mathreu, http://www.ams.org/opportunities
(For that last one, click the box to filter by Research Experience for Undergraduates.) Many of these summer research programs go by the name Research Experience for Undergraduates (REU). Visit the link above for a listing of these programs. There are often over one hundred REU programs at universities throughout the United States. The National Science Foundation, which funds REU sites, describes a typical REU experience as follows. “An REU site consists of a group of ten or so undergraduates who work in the research programs of the host institution. Each student is associated with a specific research project, where he/she work closely with the faculty and other researchers. Students are granted stipends and, in many cases, assistance with housing and travel.” Application deadlines for REUs usually occur in February or March.
- The Cal State LA U-Rise program: aims to significantly increase the participation of underrepresented groups in leadership positions within biomedical research. The project will identify and develop creativity and intellectual talents of motivated trainees from groups traditionally underrepresented in biomedical research and strengthen their academic and research credentials through 1) mentored research, 2) entering research curriculum, 3) faculty mentor training in inclusivity, and 4) building scientific identity with community, so that they are highly competitive for admission to and success in top Ph.D. programs."
- SURF at CalTech: https://sfp.caltech.edu/undergraduate-research/programs/surf Summer Undergraduate Research Fellowships program
- WAVE at CalTech: https://sfp.caltech.edu/undergraduate-research/programs/wavefellows The WAVE Fellows program provides support for talented undergraduates intent on pursuing a Ph.D. to conduct a 10-week summer research project at Caltech. The WAVE Fellows program aims to foster diversity by increasing the participation of underrepresented students in science and engineering Ph.D. programs and to make Caltech's programs more visible and accessible to students not traditionally exposed to Caltech. The program is extended, but not limited, to underrepresented minorities, women, first-generation college students, geographically underrepresented students, educationally and financially disadvantaged students, and students with disabilities.
- MSRI Undergraduate Program (MSRI-UP). “The MSRI Undergraduate Program (MSRI-UP) is a comprehensive summer program designed for undergraduate students who have completed two years of university-level mathematics courses and would like to conduct research in the mathematical sciences.”
- LSAMP. "The primary goal of LSAMP is to provide professional development in STEM careers and enhance LSAMP students’ competitiveness for and success in graduate programs and professional careers in STEM. We engage students in STEM research and other professional development activities in a supportive community."
- Causeway (The Causeway Postbaccalaureate Program is a yearlong experience that seeks to increase the number of graduate students in the mathematical sciences from historically underrepresented groups.)
- The Penn State math PhD program offers an optional extra year to receive additional training. Contact their department for more information.
- Iowa State Post-baccalaureate Certificate in Mathematics. "In this program, students will take four graduate courses, serve as teaching assistants in undergraduate courses, get ample mentoring from the department’s faculty and graduate students, attend conferences, and receive professional development to aid in applying to graduate programs (or jobs, if appropriate)."
- Wells Fargo's Quantitative Analytics Program. "The rotational Quantitative Analytics program is designed to provide you with the opportunity to gain comprehensive professional and industry experience that prepares you to develop, implement, calibrate, and validate various analytical models."
- The National Data Mine Network: "An NSF-funded grant in collaboration with the American Statistical Association to enable MSIs’ undergraduates to learn data science through research or industry projects. Provides $4500 in monthly research stipends ($500/month) plus up to $500 for conference travel to 100 students annually. Projects run throughout the 9-month academic year (August-April) with access to support, training, materials/tools , and high-performance computing from Purdue. Students work on projects with research mentors or Industry Partners through the Corporate Partners Program. Currently, there are 80+ corporate partner projects with plans to expand." Students and faculty can apply here to participate: https://datamine.purdue.edu/apply/
- NNSA-MSIIP. "The National Nuclear Security Administration Minority Serving Institutions Internship Program (NNSA-MSIIP) provides paid opportunities for undergraduate and graduate students enrolled at Minority Serving Institutions."
- GROW. "The GROW conference is intended to provide support and encouragement to undergraduate students considering going to graduate school in the mathematical sciences and to help build a profession in which gender identity is not a barrier to participation."
- The Bridge for Pursuing Higher Degrees program "invites prospective graduate students to join us for a free and online mentoring program designed to help you prepare for graduate school—from the application process, to becoming a grad student, to thriving on your way to your graduate degree."