My list (necessarily limited to what I know about,
have on my bookshelf, and have studied at least
significantly) of mathematical masterpieces? Sure:

Halmos, Finite Dimensional Vector Spaces

He wrote this in 1942 as an assistant to John von
Neumann at the Institute for Advanced Study, and the
book is baby Hilbert space. Maybe use as a second
book on linear algebra, but, if you wish and want to
try, a first book.

Rudin, Principles of Mathematical Analysis

AKA baby Rudin. Prove the theorems of calculus;
see how such math is done; learn some more material
important in the rest of mathematical analysis.

Spivak, Calculus on Manifolds

The three above were at one time the main references
for Harvard's famous Math 55.

Royden, Real Analysis

Measure theory and a start on functional analysis.
Elegant.

Rudin, Real and Complex Analysis

Rock solid, measure theory again, and more on
functional analysis. Also von Neumann's cute proof
of the Radon-Nikodym theorem. Nice treatment of
Fourier theory. Some more nice material not easy to
find elsewhere.

Neveu, Mathematical Foundations of the Calculus of
Probability

A second or third book on probability. Succinct.
Elegant. My candidate for the most carefully done,
serious writing ever put on paper.

Earl A. Coddington, An Introduction to Ordinary
Differential Equations

Rock solid mathematically, nice coverage for a first
book, and also really nicely written. Read after,
say, Halmos and baby Rudin.

Luenberger, Optimization by Vector Space Techniques

Or, fun and profit via, surprise, the Hahn-Banach
theorem, Kalman filtering, high end Lagrange
multipliers, deterministic optimal control, little
things like those, solid mathematically, succinct,
at times very applicable. I suspect that one of his
theorems is the key to a high end approach to the
usually mysterious principle of least action in
physics, etc. Reading the Hahn-Banach theorem is
just a nice evening in Royden or Rudin R&CA, but seeing
the astounding consequences for a lot of applied
math, e.g., in parts of engineering, is not trivial
and is made easy by Luenberger. It's a lesson:
Some of pure math can be much more powerful in
applications than is easy to see at first.

John C. Oxtoby, 'Measure and Category: A Survey of
the Analogies between Topological and Measure
Spaces'

Elegant. Astounding. Some of what learn via the
Baire category theorem can shake one's intuitive
view of the real line and our 3-space. Definitely a
masterpiece. Maybe it's profound.

Bernard R. Gelbaum and John M. H. Olmsted,
Counterexamples in Analysis

When studying Rudin, Royden, etc., don't be without
this one! And it's astounding and clears up a lot.
Or, why didn't Rudin state the theorem this way?
Because that way it's not true -- see Gelbaum and
Olmstead!

There are no doubt many more masterpieces, but
these are the ones I can recommend.

But, for a good background in pure and applied math
and for doing research and making applications, more
is needed. While I can list more good sources, I
can't regard them as masterpieces. E.g., I don't
know of a masterpiece in optimization, statistics,
stochastic processes, differential geometry, partial
differential equations, or abstract algebra. Useful
texts? Yes. Maybe really good? Yes.
Masterpieces? No.