There are many different kinds of graduate programs in the mathematical sciences. In addition to "regular" master's and Ph.D. programs in mathematics, applied mathematics, and statistics, there are "professional" master's programs that prepare students for specific kinds of jobs, such as financial mathematics, actuarial science, or K-12 mathematics teaching. The advice given here will help you prepare for all of these programs, but it is most closely tailored to doctoral programs in pure or applied mathematics. If you are interested in a different kind of program, ask one of your professors for more specific advice.
Advice on courses in mathematics
Take as much math as you can. In addition to the Calculus sequence, Principles of Mathematics (MATH 266), and Introduction to Linear Algebra (MATH 315), you should try to take 7-10 upper-level math courses. These should include the 400-level sequences in algebra (MATH 411/412) and analysis (MATH 421/422).
You can still consider graduate study in mathematics even if you have not taken the algebra and analysis sequences. However, you will not be eligible for many of the highly competitive programs, and you will probably have to play catch-up when you get to graduate school.
You should seek both breadth and depth in your course work. Taking courses in several different branches of mathematics will help you decide what parts of mathematics you like best and will prepare you for graduate programs of different types. It will also provide a good background for the Graduate Record Exam in mathematics.
Taking sequences (particularly the algebra and analysis sequences) will let you study a single area in depth and will prepare you for the level at which graduate courses are taught. The faculty can help you choose courses that match your needs and prepare you for the kind of graduate program in which you are interested.
If you are interested in analysis or applied mathematics, for example, then complex variables would be a good course to take. Differential equations also play a central role in applied mathematics. If you are interested in statistics, then you should take the probability-statistics sequence. If you want to explore an area of mathematics that isn't offered at Saint Louis University, it may be possible to take relevant courses at Washington University and University of Missouri - St. Louis.
As for scheduling, you should plan to take two math courses each semester, after you finish MATH 315 (Introduction to Linear Algebra). The ideal situation is one in which you finish MATH 315 by the end of your sophomore year. That will enable you to take the algebra and/or analysis sequence in your junior year, so that you can consider taking some graduate courses during your senior year. Taking graduate courses as an undergraduate strengthens your application to graduate school.
Advice on courses outside mathematics
Taking a course in computer science can be useful, especially in applied mathematics. Experience with programming helps you develop your analytical skills, and some summer research programs specifically recruit students with computer skills. If you are interested in getting a Ph.D. in mathematics, take French, German, or Russian as an undergraduate. Most Ph.D. programs in mathematics require students to demonstrate a reading proficiency in French, German, or Russian. Although it is possible to learn these languages in graduate school, it is more efficient to learn one of them now.
Activities outside the classroom
Participate in at least one mathematics-related activity outside of class. These activities demonstrate your commitment to mathematics, and they are fun! They will also help you to decide whether you are interested in going to graduate school in math. Here are some examples of things that you can do:
Since most graduate students in mathematics have financial aid packages that involve teaching introductory courses, it's a good idea to get some experience explaining mathematics to others. This can take the form of tutoring or giving a talk to the Math/CS Club or at a math conference. Find out whether the graduate programs in which you are interested require the GRE Subject Test in Mathematics. If they do, familiarize yourself with the topics covered on the test and study carefully for the test. Do not take the test "cold": half of it is on calculus, and your calculus may be rusty by the time you take the test. Note that you probably want to sign up for the November subject test and it has a September registration deadline.
A final word of advice
Seek advice from many sources. Graduate programs in mathematics vary widely in what they include and what they expect from applicants. Seek the advice of at least three faculty members about what courses you should take and what other activities you should be involved in.