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.

Very nice!

Thank you very much!