The NSF REU program funds experiences where a group of undergraduates from different institutions gather and do research on a math topic for ten weeks over the summer; typically the students also receive a stipend of several thousand dollars.
A list of programs is available here. Deadlines are typically in February and March. (Most NSF programs are only available to US permanent residents).
There are also a number of summer programs listed at mathprograms.org. You can look around without making an account; the list of undergraduate programs seems to be here.
The AMS has an info page rounding up opportunites for summer internships and other related programs.
The MAA also has a page of internship opportunities.
SIAM has a page on internships.
The AMS has a page on careers here.
The MAA has a roundup pages on careers here.
The AWM has a careers page here.
SIAM has a careers page here.
You can also see some good information on careers from non-math organizations. I thought this was a decent roundup on the career market for math majors.
Less obviously, a math major is a surprisingly good preparation for the LSAT and law school.
Fewer than 800 law school applicants majored in math, but more than 87 percent of those applicants were admitted to law school. Their average LSAT score was higher than 161, and their average GPA was 3.54. Notably, this group of applicants had the highest average LSAT scores than students with any other major. While the American Bar Association does not recommend that pre-law students select a specific major, the organization recommends a strong foundation in math courses. It specifically stresses that applicants should have mastery of math at the pre-calculus level at a minimum as well as an understanding of financial math concepts.