There are different types of math majors, as well as different types of careers for math majors.
Most math and applied math majors require Calculus III and Linear Algebra. If it is not required, I would recommend taking basic Probability and Statistics classes. I would also recommend the math classes taken by engineers, basic Differential Equations and Engineering Mathematics. Not everyone would agree with that, and you don’t need those if going into business math.
In the 1950s, most math taught in colleges was theoretical math. There has been a movement away from that, because it is not useful in the real world and for careers. I would strongly recommend against studying theoretical math. Often liberal arts colleges offer mostly theoretical math for majors. Some universities have two math departments, one theoretical and one applied. Typical theoretical math classes are Modern Algebra, Analysis, Topology, Combinatorics, Graph Theory, and Number Theory. Number Theory is sort of interesting and the theoretical math version of Differential Equations might be reasonable to take.
In general, math applied to business is more “marketable” than math applied to engineering is more marketable than theoretical math. You generally want to look for an undergraduate school with well set up applied math options.
Graduate school in theoretical math has its problems. As a credential, you have a math degree and a graduate degree. However, most of the people studying it are really smart and there is not much available in terms of career. It is generally better than a humanities or anthropology graduate degree.
If you pursue a PhD in a mathematical field, you should be aware that about half of the people who complete the course work do not get PhDs. To have a good academic career you will need good publications. The more marketable the field and the better your graduate school, the better schools you can teach at initially with a PhD. A large portion of those who get academic jobs do not get tenure initially. Then you may need to settle for a lower rank school or a career outside academia.
Applied mathematics can also be fairly theoretical with proofs etc. There is a tendency for theoretical mathematicians to get academic positions in more applied areas making those theoretical.
Math professors have a reputation for copying some long theorem into their notes and then on to the board. In advanced classes, it may take 2 days to prove the theorem.
Many Applied Math majors, particularly at good schools, are interested in applying to MBA programs. This works well with math oriented towards probability and statistics and operation research. At top MBA programs, the students talk about “poets” and “quants”. The “poets” are humanities majors who either planned on going the MBA or law school or figured they needed something else to get a good job. The “quants” are mathematically oriented. Many top universities and liberal arts colleges still do not offer business classes. I would recommend taking some business and/or economics classes if planning on an MBA, but it is not required. Taking some humanities is not at all required, but it helps to speak and write well.
Anyone with a math major can teach high school, as there is a big demand for that. Part of the reason is the pay the same or close to elementary school teachers. You need to take a bunch of education courses, which many math majors might not be comfortable. Sometimes, they have accelerated programs for that and you may not need those for private schools. You need to control the classroom etc., which many math majors are not comfortable with. There are many math teachers who cannot teach higher level classes like calculus or honors classes. Generally, their math skills are not top level. Math education degrees tend to be offered by second tier state colleges. However, you usually need to start with easier classes and get experience even if you think you know the math. It is a reasonable option for someone good in math. You may be able to teach higher end classes in fancier districts or at private schools. You may be able to get good rates for tutoring high end subjects. High school math teachers tend to have the reputation of being rigid, by the book, and by the numbers.
Actuaries are mostly math majors. That requires about 10 exams you take while working. The salary when finished the exams is close to $200K / year. There are many actuaries in high management positions in insurance companies. Actuaries are partly paid well for a sophisticated understanding of the finances of insurance companies etc.
There are colleges with actuarial science programs, which specifically prepare you for the exams. Those are generally somewhat competitive but not top tier schools. A math and business major is also good background for that, but a straight math major will do.
The exams may seem interesting, but the work is probably dull. Actuaries have a reputation for being even more boring than accountants, lacking in personality and very cold. It probably takes a certain personality or lack thereof to work with mortality tables.
Not everyone who passes the first exams gets an actuarial job. Although maybe dull, you need good communications skills. It deals with important financial analysis for the business of the insurance companies etc. However, it is easy to change assumptions and get whatever desired results. Casualty actuarial work deals a lot with rate filings with state insurance departments. They generally want good talkers etc. to handle interactions which may not be entirely what they seem.
Computer science may seem mathematical and have good career prospects etc. However, some it is theoretical and math related because theoretical mathematicians could get academia positions teaching it. Computer science is good as a credential, but what you learn may not be that useful or respected. The computer classes from an engineering perspective are more solid.
There are areas in computer programming for which math is much used, such as graphics and statistical programming, but you will not automatically get into them by having a math degree or math background.
There are some could programs, particularly at engineering school about math applied to engineering. The only problem is that graduate school in engineering is often math-oriented more than engineering work is, so there may not be a great need for that sort of applied math. That being said, it is a solid degree, which can be used in programming engineering applications, etc.
Tutoring can be done full or part time. Generally, you will need to work up to more advanced subject and honors classes unless you have a strong academic background or teaching experience. Most high school teachers of advanced or honors classes do some tutoring and it is a way to supplement teaching salaries.
Tutoring full time can be tricky is terms of getting enough good business. Many tutoring services charge a high markup and hire college students. A standard business model is Ivy tutoring services, which market students with good academics as top tutors. Those tutors can generally tutor both the math and verbal SATs for students going for top scores.
The higher rates are usually for the more advanced subjects and math SATs. Honors math classes at private schools can be extremely difficult, so require expert tutors, and the parents can afford to pay. There are generally high rates tutoring private school classes in NYC.
Most of this I have had experience with or no people who have had experience with.
