In Experimental and computational mathematics: Selected writings Jonathan Borwein states the following about The psychology of invention
in the mathematical field by Jacques Hadamard
[The psychology of invention
in the mathematical field is] a book that still rewards close
inspection
[Hadamard] was perhaps the greatest mathematician
to think deeply and seriously about cognition in mathematics
Borwein also gives an analysis of A Mathematician's Apology. Borwein does state that some of the idea's expressed throughout the book are outdated, but nevertheless quotes Hardy's view on mathematical beauty multiple time in his paper.
The Apology is a spirited defense of beauty over utility: ‘‘Beauty is the first test. There is no permanent place in the world for ugly
mathematics.’’
In a publication of Journal of Recreational Mathematics Charles Ashbacher wrote the following about A Course of Pure Mathematics by G. H. Hardy
"Although the sequence of the presentation of the fundamentals of mathematics has changed over the last century, the substance has not. There is no greater evidence of this fact than this classic work by Hardy, which could be used without alteration or additional explanation as a text in modern college mathematics courses... The mathematical influence of G. H. Hardy over mathematical education was and remains strong, as can be seen by reading this masterpiece."
The user Nathan provides the following quote by Herman Weyl in an article for the Mathematical Review concerning G. Polya's How to Solve It (added by permission):
This Elementary textbook on heuristic reasoning, shows anew how keen its author is on questions of method and the formulation of methodological principles. Exposition and illustrative material are of a disarmingly elementary character, but very carefully thought out and selected.
E. T. Bell also praised How to Solve It, having written in Mathematical Monthly that
Every prospective teacher should read it. In particular, graduate
students will find it invaluable. The traditional mathematics
professor who reads a paper before one of the Mathematical Societies
might also learn something from the book: 'He writes a, he says b, he
means c; but it should be d.
Continuing the trend of Polya's works, Mathematics and Plausible Reasoning is key. In The Mathematics Teacher Bruce E. Meserve defends the book as
...a forceful argument for the teaching of intelligent guessing as well as proving. . . . There are also very readable and enjoyable discussions of such concepts as the isoperimetric problem and 'chance, the ever-present rival of conjecture.'
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Naoki Saito, Professor of Mathematics Department of UC Davis, has published a list of his recommended books. Unfortunately, the author does not provide a quote for each (in order to save space in the already extremely long list). However, I have condensed a number of my personal favorites he recommends that fit your criteria (except for quotes). I have placed an asterisk by books that lean more towards mathematical physics, as I feel these should be included but should be noted to not be pure mathematics.
E. T. Bell: Men of Mathematics
Max Born: Principles of Optics*
R. Courant & David Hilbert: Methods of Mathematical Physics*
F. R. Gantmacher & Mark Krein: Oscillation Matrices and Kernels and
Small Vibrations of Mechanical Systems
R. P. Feynman: Lectures on Physics*
F. R. Gantmacher: The Theory of Matrices
P. R. Garabedian: Partial Differential Equations
G. H. Hardy: A Course of Pure Mathematics
G. H. Hardy: Divergent Series
G. H. Hardy, J. E. Littlewood, & G. Pólya: Inequalities
G. H. Hardy & E. M. Wright: An Introduction to the Theory of Numbers
H. Helmholtz: On the Sensations of Tone*
H. Helmholtz: Treatise on Physiological Optics*
T. Kato: Perturbation Theory for Linear Operators
O. Kellogg: Foundations of Potential Theory
C. Lanczos: Applied Analysis
C. Lanczos: Linear Differential Operators
C. Lanczos: Discourse on Fourier Series
P. M. Morse & H. Feshbach: Methods of Theoretical Physics*
G. Pólya: Mathematics and Plausible Reasoning
J. W. S. Rayleigh: The Theory of Sound*
W. Rudin: Real & Complex Analysis
V. I. Smirnov: A Course in Higher Mathematics
E. C. Titchmarsh: The Theory of Functions
G. N. Watson: A Treatise on the Theory of Bessel Functions
E. T. Whittaker & G. N. Watson: A Course of Modern Analysis
K. Yosida: Functional Analysis
A. Zygmund: Trigonometric Series
Best Answer
Ravi Vakil's The Rising Sea: Foundations of Algebraic Geometry is an excellent textbook on (you've guessed it) foundations of algebraic geometry, very nicely mixing rigorous proofs, intuitive explanations and illustrations. It is available for free on author's website, and the newest version (along with all other versions) is always available here:
http://math.stanford.edu/~vakil/216blog/index.html
There is also Fulton's book Algebraic Curves: An Introduction to Algebraic Geometry is another nice book whose latest edition has been published for free by the author. It gives an elementary introduction to the theory of algebraic curves, up to desingularization and Riemann-Roch theorem.
http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf
Both books require some familiarity with abstract algebra, but assume no prior contact with algebraic geometry.