Adding descriptions when I can - rather busy at the moment with exam preparation. I really shouldn't be working on this at all, to be honest.
My purpose here is to present the best of the best that textbooks have to offer: the most rigorous, the most difficult, and the most rewarding.
Difficulty levels:
For the student new to the subject of calculus, Kline is not only extraordinarily affordable but also quite accessible. Lang is also an option if one wishes for a more rigorous introduction, although Lang's books are somewhat dated and do not necessarily cover all the topics in a modern calculus class; nevertheless, his exposition in his calculus texts is wonderfully clear.
Spivak, Apostol, and Courant are reserved for either the talented honors student or the student wishing to cement his understanding of the fundamentals of calculus. Spivak is perhaps the most rigorous of the three with regards to proofs, while Apostol's strength is his thoroughness and straightforward presentation. Courant - both the original and the rewrite with John - takes a classic but also more intuitive approach (although I have heard that the rewrite loses some of the charm of the original, even while improving in some other minor areas). Any of them will prepare a student adequately for analysis and other fields. (I am admittedly somewhat partial to the original Courant books due in no small part to their relative inexpensiveness compared to the other calculus classics.)
Schey and Hubbard give decent treatments of vector calculus - the material should be covered fairly decently by Spivak/Apostol/Courant, but I imagine they make for decent supplements.
As far as ordinary differential equations go, Tenenbaum gives a fairly comprehensive overview of common methods for solving ODEs. Arnold presents a more theoretical understanding of the topic; both have high-quality exposition and a fair number of exercises.
Farlow's PDE book is suitable for those who need merely a functional understanding of the topic. Evans is excellent as a textbook; the exposition is clear and uncluttered, and nonlinear PDEs are covered thoroughly. The monumental Taylor three-book series is an exceptional reference and essentially contains everything related to differential equations. Similarly, Hörmander is the definitive reference for linear PDEs - truly encyclopedic.
Shilov gives a nice overview of introductory linear algebra, while Axler gives a more theoretical treatment of the subject. Halmos also makes for an excellent second book in linear algebra, presenting the material in a generalized fashion with elegant exposition and rigorous proofs: an excellent companion to Axler. Lang, as always, is quite dry, while Greub's encyclopedic text is an excellent reference; it also serves as a sequel to Halmos, expounding greatly upon topics that Halmos mentioned only briefly.
Pinter's text is renowned for being quite accessible; the exposition is very readable and should pose no difficulties. Herstein and D&F present the topic in a more rigorous manner; D&F is better as a reference, while Herstein's exposition is exceptional and the exercises develop a very thorough understanding of abstract algebra. D&F, in contrast, while good, has somewhat overcomplicated and convoluted exposition (but still remains a quality text - simply one more suited for reference purposes). Artin presents algebra in a somewhat unconventional manner, particularly with an emphasis on nontrivial computational exercises and examples; it may prove rather inconvenient for self-learners, as the proofs are quite concise, but offers a very intuitive treatment of the subject.
The Lang book is, as always, dependent upon personal preference. Jacobson is rather dense, but covers a great number of topics in a comprehensive, rigorous manner.
For the reader who has had a somewhat lacking background with regards to calculus or perhaps finds himself or herself struggling with 'Baby Rudin', as it is commonly called, Shilov may prove to be of utility. Gelbaum and Olmsted proves to be a useful companion to the student of analysis at any level, pointing out peculiar eccentricities that often go overlooked. However, for the student with a solid grounding in mathematical rigor, calculus, and linear algebra, it is appropriate to begin with Rudin's text.
It is generally agreed that Rudin's Principles of Mathematical Analysis serves as an excellent introduction to the field of analysis. The exercises are challenging and the exposition is very clean. Apostol is noticeably drier; the text has a bit of complex analysis, and is somewhat more accessible, as it caters to readers with a lower level of mathematical maturity - regardless, a decent text. The other medium-level texts are not particularly necessary; they do give a more rigorous, theoretical treatment of analysis. Hewitt and Stromberg as well as Dieudonné may prove to be worthwhile; both are quite formal and present many difficult exercises for the reader.
Generally speaking, though, one may advance straight to Rudin's Real and Complex Analysis after completion of his Principles. Lang's book is dry as usual; we see here his Bourbakian roots quite clearly, and whether or not the generality and modernity of Lang's text is appreciated may vary between readers.
(The difficulty of Rudin differs here from its difficulty in the previous section not to say that the book's second half is easier but rather as a more accurate indication of its relative standing among the other texts here.) Ahlfors is a classic of complex analysis; while dense, it presents the material in a fairly understandable manner. The second half of Rudin deals primarily with complex analysis; the exposition is clear and the exercises difficult. Narasimhan covers a great number of topics and progresses at a rapid pace; it is excellent as a reference and as a source of exercises, but may prove frustrating as a first exposure to complex analysis. Lang is comprehensive and dry - rather difficult to read, but undoubtedly worthwhile.
Andersson's work on advanced complex analysis is an excellent follow-up after one has worked through an introductory text. Similarly, Gunning and Rossi does well in its treatment of the theory of several complex variables, which is further expounded upon by Hörmander, whose text yet again serves quite well as a reference and treats the topic with impressive generality and rapidity. Jost is a very accessible introduction to Riemann surfaces, while Narasimhan is a great deal more challenging and preassumes quite a bit of prior exposure to Riemann surfaces.
Kreyszig gives a fairly understandable (albeit application-oriented) introduction to functional analysis. It has quite a lot of explanation and reads somewhat like a physics text. Generally, though, going straight into Rudin or Kadison and Ringrose should not prove to be excessively difficult, although the latter is known for its exceptionally difficult exercises. Dunford and Schwartz is regarded as a classic in the field; difficult to read compared to modern mathematics, but an invaluable reference nevertheless.
Ireland and Hardy are both standard texts for introductory number theory; both have clear exposition and are essential for any student of number theory. Do note that Ireland assumes some familiarity with basic abstract algebra. Apostol, while dry as usual, covers elementary number theory in a concise manner and includes many relevant exercises. Although Apostol does not presume any prior knowledge of number theory, it does progress at a rapid rate, so exposure to the topic may be helpful.
Fröhlich and Taylor covers a prodigious number of topics while Narkiewicz is an exceptional reference, containing pages of historical notes and unsolved problems. The Silverman texts on elliptic curves are essentially the seminal works regarding that particular topic, while Koblitz offers a fairly readable treatment of p-adics.
Basic introductory topology is covered fairly well by all of the easy-classified texts. Willard's exposition is superb, while Munkres is a timeless classic; Steen and Seebach makes for an excellent companion to any introductory text. The topic can be approached with any of the introductory books to anyone with a rudimentary grasp of calculus, even.
Guillemin and Pollack is an exceptional first course in differential topology, requiring little more than some multi-variable calculus, linear algebra, and point-set algebra: the exposition is intuitive and visual, leading to a comprehensive understanding of the fundamentals. Milnor is a similarly comfortable introduction to the topic of differential topology; Hirsch, while somewhat technical, has intriguing exposition and serves well as a second textbook. The Serge Lang text is dry yet modern in the Bourbakian style; its utility depends greatly upon personal preference.
Organized similarly to Spanier's encyclopedic text, Rotman's introductory book serves concise yet illuminating exposition and presents introductory material in a very comprehensible fashion. Bott and Tu is an exceptional treatment of algebraic topology; the exposition is clear and the treatment of de Rham cohomology and spectral sequences is superb: the text is certainly not an introduction, but serves well as a second or third course in algebraic topology. Fulton's exposition is also laudable; the material is presented in a slightly less rigorous fashion, serving as an excellent supplement to any introductory text. Spanier makes for an atrocious instructional textbook but serves as a wonderful reference and source of difficult exercises.
Euclid's Elements is a timeless classic and outlines the development of the fundamentals of classical geometry. Coxeter's two books develop the subject of geometry further, while Hartshorne is an excellent companion to Elements.
Spivak's treatment of this topic is decent; however, no exercises are provided. Do Carmo, on the other hand, has an abundance of challenging theoretical exercises - very little in the way of drill. Kobayashi and Nomizu is a fairly poor choice of textbook, particularly for self-study, but is an excellent reference - not, however, written with pedagogical or autodidactical instruction in mind.
These are the standard texts of probability theory. Feller, in his second volume, begins to delve more and more into measure theory and its applications to the field of probability; both volumes are very well-written. Drake presents some practical considerations and is also an excellent introductory text.
The Stanley volumes are, currently, the standard and seminal works on combinatorics and counting. Chartrand gives a thorough overview of the subfield of graph theory, while Bollobas expands upon the basics and gives a more modern presentation. Lovász is, of course, a combinatorial classic, with a plethora of problems spanning every difficulty level.
Larson and Zeitz both give a fairly decent overview of basic problem-solving strategies, while Engel, Mahajan, and Pólya present more complex and difficult techniques as well as more challenging exercises. Pólya's Problems and Theorems in Analysis books will present a challenge for even the most experienced problem-solver, drawing from a wide array of mathematical topics (complete solutions are provided). Lovász is a Hungarian classic, possessing a truly astounding variety of combinatorial problems, and Arnold's problems are notoriously difficult.
Generally speaking, working through these problems is a worthwhile and educational pursuit - quite recommended for anyone seeking to perform well in competitive mathematics.
Schwarz and Fleisch do not quite stand on their own as instructional texts; rather, they make for superb supplements to the classic introductory texts. Griffiths and Purcell are the traditional undergraduate texts for this subject, and should accordingly pose a bit more of a challenge; finally, Jackson is almost exclusively a graduate-level text, and is renowned for its difficulty and rigor. Jackson should be attempted by those with a solid grounding in electrodynamics who wish to attain mastery of the field and will require prior knowledge of many mathematical tools for full comprehension.