Many Mathematics Departments and Universities have summer programs called "Research Experiences for Undergraduates," or REUs for short. They tend to be 6-8 week summer programs aimed at helping undergraduates become involved in mathematical research. If you are considering going to graduate school, these research experience programs are an excellent way to "test-drive" graduate school. Many such programs are funded by the National Science Foundation and provide travel expenses and stipends, at least to US citizens and permanent residents. The NSF maintains a directory of currently operating REUS:

https://www.nsf.gov/crssprgm/reu/list_result.jsp?unitid=5044

Although details vary by individual site, applications tend to be due in February each year.

Another good resource for searching for mathematically oriented research experiences and summer programs is:

https://www.mathprograms.org