I'm just wondering, but what math classes are the ones that all math majors ought to take?
The standard classes are usually the calculus sequence, linear algebra, a semester or two of analysis, a semester or two of algebra, differential equations, and a couple electives.
Usually one would take the calculus sequence + a class on Ordinary Differential Equations + a class on elementary/computation based Linear Algebra during their first 2 or 3 semesters. Then one would take a "transition to higher math" class where one learn proofs and such. Beyond that point, ideally one would take at least a semester of Real Analysis, Complex Analysis, Abstract Algebra and Topology each. A proof-based Linear Algebra class is also important and a lot of people take that too. Other than those, for your electives, one choose a couple of classes from say, Differential Geometry, Partial Differential Equations, Stochastic Processes, Probability etc. Oh and I think a lot of schools these days also have a programming/computational class as a requirement so most math majors would also take some introductory programming class.
A lot depends on whether you are a pure or an applied mathematics major.
In either case, classes like calculus, diffy eq and linear algebra can not be avoided.
If you're a pure mathematics major, then you will also need classes like abstract algebra, real analysis, complex analysis and topology. Other classes depend on your university and your interests.
If you're applied, then you might take things like statistics, numerical analysis, more differential equations.
What if you're not sure which direction you want to go after the Bachelor's level, and want to get exposed to as many fields as possible? What classes should you take, then?
If you are not sure, then you should probably take a little bit of both, and that "little bit" depends on how flexible your schedule/program is (e.g. some math departments can let you take whatever you want and still graduate, whereas the others want to take you specific number of courses from either pure or applied. This is something you should talk to your academic advisor).
In any case, though, if you are interested in both pure and applied, I would definitely consider taking real and complex analysis ASAP--every pure math majors ought to know rigorous foundations of calculus, and these courses can provide you theoretical background in many areas of applied mathematics (be it PDE, stochastic modeling, numerical analysis, and etc...). Also, take as many linear algebra as you can, whether it is a theoretical or applied linear algebra. A solid understanding of linear algebra is essential in any area of mathematics, be it pure or applied.
You could get a healthy mixture of both pure and applied. There is nothing wrong with that (I would even recommend it). Are there things you are interested in right now? I'm sure that if you chose math as a major, that there must be things in math that you would like to see more of? Choosing classes based on what you would like to see is always a good idea.
Anyway, the following classes are things I consider to be fundamental:
Calculus I, II, III
Intro to Proofs (I'm not a fan of a class like this, but I guess that taking it can't hurt, although I did perfectly fine without it)
Linear Algebra (try to do a proof-based version of Linear Algebra)
Probability Theory/Statistics (try to take a mathematical statistics class that uses measure theory, although this should not be your first statistics class)
Numerical Analysis/Linear algebra
Right now, I'm just sort of really uncertain of things, and I thought that doing Math would open the most doors, since I thought I could go to math graduate school, or statistics, or actuarial work, and I also read that taking higher-level math is essential for econ grad school, and I thought that if I found out liked engineering, I could just get a Masters in it after getting a bachelor's in math since I saw on some websites that that was possible, and I also have some interest in computer science, or maybe even physics... with physics I admit I have a bit more of a fascination in the tools used to solve difficult physics problems, like the calculus of variations and things like that, though. But all of the plans I have center on getting at a minimum a bachelor's degree in math, if not a double major with that and something else.
I also have a bit of an interest in operations research, but the only thing I really know about it is that it's a reasonably profitable field and that it uses a lot of discrete math and probability.
In other words, I feel really indecisive now.
That's okay. You shouldn't have to make a decision right this very second. Declare mathematics as your major and after taking a few courses, you'll find what interests you. No matter what path you take (pure, applied, discrete), you'll have to take the same courses your first year or two anyways, assuming you're going to school in the US. It usually isn't until your Junior year when you'll be required to pursue a particular "specialization." Relax, you've got time to figure out where you want to go.
My undergrad institution required applied math majors to take (upper division) linear algebra, real analysis, and either a 3 quarter sequence on probability, statistics or numerical analysis. Then, you pick out electives.
Pure math required real analysis, algebra and then electives.
My advice is to spend a lot of time in the university library. Try browsing for books which sound interesting, look at the contents and read the first section.
In the best case you find an interesting topic that you are able to understand. You can start reading on the side or keep it in mind for later. If it's beyond your level, no worries. At least you learned about a new field, so it wasn't in vain. You can come back later if it sounds interesting.
At least this worked for me. I was able to discover my interests for mathematical physics and topology in this way.
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