I'm still a student, but the same books keep getting named by my tutors (Rudin, Royden).
I've read Baby Rudin and begun Royden though I'm unsure if there are other books that I "should" be working on if I want to study beyond Masters. I'm not there yet as I'm on a four year course and had a gap year between Years 3 and 4.
Please recommend for Algebra, Linear Algebra and Categories – Analysis, Set Theory, Measure theory (an area I have seen too little books dedicated for).
E.g. Spivak is very good for self learning basic real analysis, but Rudin really cuts to the heart.
Best Answer
EDIT: I now think that this list is long enough that I shall be maintaining it over time--updating it whenever I use a new book/learn a new subject. While every suggestion below should be taken with a grain of salt--I will say that I spend a huge amount of time sifting through books to find the ones that conform best to my (and hopefully your!) learning style.
Here is my two cents (for whatever that's worth). I tried to include all the topics I could imagine you could want to know at this point. I hope I picked the right level of difficult. Feel absolutely free to ask my specific opinion about any book.
Basic Analysis: Rudin--Apostol
Measure Theory: Royden (only if you get the newest fourth edition)--Folland
General Algebra: D&F--Rotman--Lang--Grillet
Finite Group Theory: Isaacs-- Kurzweil
General Group Theory: Robinson--Rotman
Ring Theory: T.Y. Lam-- times two
Commutative Algebra: Eisenbud--A&M--Reid
Homological Algebra: Weibel--Rotman--Vermani
Category Theory: Mac Lane--Adamek et. al--Berrick et. al--Awodey--Mitchell
Linear Algebra: Roman--Hoffman and Kunze--Golan
Field Theory: Morandi--Roman
Complex Analysis: Ahlfors--Cartan--Freitag
Riemann Surfaces: Varolin(great first read, can be a little sloppy though)--Freitag(overall great book for a second course in complex analysis!)--Forster(a little more old school, and with a slightly more algebraic bend then a differential geometric one)--Donaldson
SCV: Gunning et. al--Ebeling
Point-set Topology: Munkres--Steen et. al--Kelley
Differential Topology: Pollack et. al--Milnor--Lee
Algebraic Topology: Bredon--May-- Bott and Tu (great, great book)--Rotman--Massey--Tom Dieck
Differential Geometry: Do Carmo--Spivak--Jost--Lee
Representation Theory of Finite Groups: Serre--Steinberg--Liebeck--Isaacs
General Representation Theory: Fulton and Harris--Humphreys--Hall
Representation Theory of Compact Groups: Tom Dieck et. al--Sepanski
(Linear) Algebraic Groups: Springer--Humphreys
"Elementary" Number Theory: Niven et. al--Ireland et. al
Algebraic Number Theory: Ash--Lorenzini--Neukirch--Marcus--Washington
Fourier Analysis--Katznelson
Modular Forms: Diamond and Shurman--Stein
Local Fields:
Class Field Theory:
Metric Groups: Markley
Algebraic Geometry: Reid--Shafarevich--Hartshorne--Griffiths and Harris--Mumford