My way to Princeton

Every young mathematicians thinks of Princeton as a hub of mathematics.Princeton has no doubt for the last century produced very successful mathematicians.Its also my dream of getting in Princetion as a Graduate student. Followings things would lead me there:

Have a High cGPA.

Have good Mathematical background. Take more advanced and proof oriented courses. Stay up at nights.

Make good relationships with your professors.

Good English Speaking skills.

Good GRE.

Publish at least some papers in Algebraic Geometry. which is my field of interest.

Participate in Mathematics Competitions international and National.

Go on seminars internationally.

Extra things to do during my undergrad

An introduction to the theory of numbers G.H.Hardy
18.104 Seminars in Analysis
18.212 Algebraic combinatorics

My Plans for Physics

Mechanics
Classical Mechanics II
Electricity and Magnetism
Relativity
Waves and Oscillations
Electromagnetism 2(Electrodynamics)
General Relativity
Quantum Mechanics I
Quantum Mechanics II
Quantum Mechanics III
Quantum Theory
Quantum Field Theory I
Quantum Field Theory II
Quantum Field Theory III
Effective Field Theory; Strong Interactions (Quantum Chromodynamics)
String Theory

Way to choose books for a particular Course

MIT ocw books
Ucla Books

Modern Algebra

Algebra: Michael,Artin

My plan for Undergraduate:

Semester 2:
18.01: Calculus I                                                     (PreCalculus skills)
18.02: Calculus II                                                    (18.01)
18.03: Differential Equations:                             (18.02)
Summer:

18.700: Linear Algebra                                           (18.02)
18.703: Modern Algebra                                        (18.02)
18.100A: Real Analysis                                           (18.02)
18.034: Hons Differential Equations                  (18.02)
(Do all above at the same time)
18.950: Differential Geometry                              (18.700,18.100A)

Semester 3:
18.112: Functions of a Complex Variable                 (18.700,18.100A)
18.900: Geometry and Topology in the plane         (18.700,18.100A)
18.950: Algebra I                                                              (18.100A)
18.101: Analysis and manifolds                                    (18.700,18.100A)
18.102: Introduction to Functional Analysis            (18.700,18.100A)
18.901: Introduction to Topology                                (18.100A)
(Do all above at the same time)

Semester 4:
18.755: Introduction to Lie Groups                             (18.700,18.100A)
18.702: Algebra II                                                             (18.701)
18.904: Seminars in Topology                                       (18.901)
18.952: Theory of Differential Forms:                        (18.700,18.101)
18.116: Riemann Surfaces                                             (18.112)
18.155: Differential analysis I                                       (18.102)
(Do all above at the same time)

Summer:
18.994: Seminars in geometry                                      (18.700,18.100A)
18.905: Algebraic Topology I                                         (18.901,18.701)
18.965: Geometry of Manifolds I                                  (18.101,18.950)
18.705: Commutative Algebra                                      (18.702)
18.782: Introduction to Arithmetic Geometry         (18.702)

Semester 5:
18.715: Introduction to Representation Theory      (18.702 or 18.703)
18.721: Introduction to Algebraic Geometry             (18.702,18.901)
18.906: Algebraic Topology II                                          (18.905)

Semester 6:
18.966: Geometry of Manifolds II                                   (18.965)
Advanced Algebraic Geometry                                        (18.721)

Summer:
Research Paper Published

Real Analysis:

Resources for Real Analysis:
Analysis Terence Tao Vol.I & Vol.II
Principles of Mathematical Analysis (Walter Rudin)
K.A. Ross, Elementary Analysis: The Theory of Calculus, 2nd Ed.  
Introduction to Analysis: Arthur Mattuck
(https://www.math.ucla.edu/ugrad/courses/math/131A)
More Advanced Texts: Real Analysis, Elias.M.Stein (see http://www.math.ucla.edu/~tao/245a.1.10f/)
(Prerequisites: Math 121: Introduction to Topology, 131A: Real Analysis, 131B: Real Analysis)