Take the standard path.

High School

Algebra I

Plane geometry (with emphasis on proofs)

Algebra II

Trigonometry

Solid geometry (if can get a course in it --
terrific for intuition and techniques
in 3D)

College

Analytic geometry (conic sections)

Calculus I and II

Linear algebra

Linear algebra II, Halmos, Finite Dimensional
Vector Spaces (baby version of Hilbert space
theory)

Advanced calculus, e.g., baby Rudin, Principles
of Mathematical Analysis -- nice treatment
of Fourier series, good for signals in electronic
engineering. The first chapters are
about continuity, uniform continuity, and
compactness which are the main tools used
to prove the sufficient conditions for the
Riemann integral to exist. At the end Rudin
shows that the Riemann integral exists
if and only if the function is continuous
everywhere but on a set of measure zero.
But what Rudin does there at the beginning
with metric spaces is more general than
he needs for the Riemann integral but is
important later in more general treatments
in analysis. Rudin does sequences and
series because they are standard ways to
define and work with some of the important
special functions, especially the exponential
and sine and cosine further on in the book.
The material in the back on exterior algebra is for people
interested in differential geometry, especially
for relativity theory.

Ordinary differential equations, e.g.,
Coddington, a beautifully written book,
Coddington and Levinson is
much more advanced) -- now can do basic
AC circuit theory like eating ice cream.

Advanced calculus from one or several more traditional
books, e.g., the old MIT favorite Hildebrand,
Advanced Calculus for Applications,
Fleming, Functions of Several Variables,
Buck, Advanced Calculus -- can now look
at Maxwell's equations and understand at least
the math. And can work with the gradient
for steepest descent in the maximum likelihood
approach to machine learning.

Maybe take a detour into differential geometry
so that can see why Rudin, Fleming, etc. do
exterior algebra, and why Halmos does multi-linear
algebra, and then will have a start
on general relativity.

Royden, Real Analysis. So will
learn measure theory, crucial for
good work in probability and stochastic
processes, and get a start on functional
analysis (vector spaces where each point
is a function -- good way to see how to use
some functions to approximate others).
Also will learn about linear operators
and, thus, get a solid foundation for
linear systems in signal processing and
more.

Rudin, Real and Complex Analysis,
at least the first, real, half.
Here will get a good start on
the Fourier transform.

Breiman, Probability -- beautifully
written, even fun to read. Measure
theory based probability. If that
is too big a step up in probability,
then take a fast pass through some
elementary treatment of probability
and statistics and then get back to
Breiman for the real stuff. Will
finally see what the heck a random
variable really is and cover the important
cases of convergence and the important
classic limit theorems. Will understand
conditioning, the Radon-Nikodym theorem
(von Neumann's proof is in Rudin, R&CA),
conditioning, the Markov assumption,
and martingales and the astounding
martingale convergence theorem and
the martingale inequality, the strongest
in mathematics. So will see that
with random variables, can look for
independence, Markov dependence, and
covariance dependence, and these
forms of dependence, common in
practice, can lead to approximation,
estimation, etc.

Now will be able to understand EE
treatments of second order stationary
stochastic processes, digital filtering,
power spectral estimation, etc.

Stochastic processes, e.g.,
Karatzas and Shreve. Brownian Motion
and Stochastic Calculus. Now can
get started on mathematical finance.

But there are many side trips available
in numerical methods, linear programming,
Lagrange multipliers (a surprisingly
general technique), integer programming
(a way to see the importance of
P versus NP), mathematical statistics,
partial differential equations,
mathematical finance, etc.

For some ice cream, Luenberger, Optimization
by Vector Space Methods or how to learn
to love the Hahn-Banach theorem and use
it to become rich, famous, and popular
with girls!