What is the typical flow chart for college mathematics courses? I know that college mathematics typically start at Calculus I (Differential Calculus) and then Calculus II (Integral Calculus). Then after Calc II, it starts to branch off a little. One option after the first two semesters of Calculus is Calculus III (Vector/Multivariable Calculus). Another option is Linear Algebra, and then the next course is Differential Equations. I know that other, more advanced math topics exist like Number Theory/Abstract Algebra, Discrete Mathematics, Partial Differential Equations, Geometry Topics and Topology. I was wondering what order the above mentioned math topics go in and what their prerequisites typically are. If anyone can provide me with a flow chart, then that would be great.
This depends heavily on the student and his/her interests. You will need almost all of these courses, but it's a bit murky after linear algebra, multivariable calculus, and differential equations. Did you learn proofs in any of those courses? If not, then maybe a proofs course. This all depends on the department that you're in and how they arrange how they teach mathematical maturity and theory.

Since you asked for a flow chart, I can give you an example of one department's flow chart for their courses. Hopefully this can give you an idea: http://www.math.uga.edu/~curr/prereqs.html

What the numbers all mean can be found here: http://www.bulletin.uga.edu/CoursesHome.aspx?Prefix=MATH

Hopefully this gives an example of what one department's "flow chart" looks like. You can clearly see there's a lot of choices to take and a huge 'choose-your-own-adventure' part to it.