If you intend to study on your own the best approach is to follow a structured sequence just like in an ordinary Math degree.... but nevertheless not forgetting that everything is interconnected and prerequisites and applications are highly nonlinear among different subjects (like remarked in some comments above). A more detailed list could be this one (each column to be learned simultaneously within the rest of topics):

**(Analysis Undergrad.)**

Calculus (one variable) -> Vector Calculus -> Functions of One Complex variable -> Measure Theory

--------------------------------> Ordinary Diff. Eq. -> Partial Diff. Eq. -> Variational Calculus -> Integral Eq.

**(Algebra & Discrete Undergrad.)**

Linear & Multilinear Algebra -> Group Theory -> Rings & Modules -> Intro to Representation Theory

Combinatorics & Graph Theory

Elementary Number Theory

**(Geometry & Topology Undergrad.)**

Affine & Euclidean Geometry -> Projective Geometry -> Differential Curves & Surfaces

----------------------------------------> Point Set Topology -> Introduction to Elementary Algebraic Topology

**(Probability & Statistics Undergrad.)**

Elementary Statistics --> Elementary Probability -> Advanced Statistical Methods

**(Analysis Grad.)**

Real Analysis -> Functional Analysis -> Complex Analysis (several variables)

Dynamical Systems (and Chaos)

Partial Differential Equations (general theory)

**(Algebra Grad.)**

Commutative Algebra -> Homological Algebra -> Category Theory

Lie Algebras -> Representation Theory

**(Geometry & Topology Grad.)**

Smooth Manifolds -> Algebraic Topology

--------------------------> Differential Topology

--------------------------> Algebraic Geometry

--------------------------> Riemannian Geometry -> Complex Geometry -> Symplectic Geometry

I do not know about advanced statistics and probability, and graduate number theory should should deal with analytic number theory and algebraic number theory with class field theory up to diophantine and arithmetic geometry.

May be you could make your own list according to your tastes looking up some course sequences and syllabus offered by good universities.