Self-Studying Analysis: Direction I have just completed a first course in real analysis (studied construction of real numbers, supremum/infimum, sequences, series, limits, continuity, differentiability and very little point set topology). 
My goal is to be better at analysis, to reach a point where proof-writing comes naturally to me and to fully understand and appreciate the beauty it entails. 
Right now, what I want to be able to do is:
1) Polish the material I've already covered. While studying, I felt there were a lot of proofs of theorems (classical ones) which I didn't fully understand and had to take them for granted(due to limited time, I had to both, understand as well as practice so to be able to attempt quizzes, assignments and mid/final).
2) Cover further material: I want to do integration (Reiman integration, Darboux integration and Lebesgue integration) as well as move on to metric spaces and cover that as well.
My questions are:


*

*How should I go about self-studying so the aforementioned goals are
met? Should I start all over again, from scratch, or if anyone can
give me a detailed plan as to studying these, that would be great!

*The other question I have is related to the importance and
understanding of the classical 'BIG' proofs. They are quite hard for
a beginner as myself(or at least that is what I think!). I feel as
if I can never approach them on my own and even understanding them
is a challenge! What should I do about this? Since the ultimate goal
is to become better and understand the material in a really good
manner, I do want to be able to absorb the proofs for such big
theorems!
Thank you!!
 A: This question's been asked many times at this site and it's really a waste of time to reinterate what I and others have said at those threads. So the best thing to do is to refer you to them: 
Teaching Introductory Real Analysis
Studying for analysis- advice
Analysis - Book recommendation with many examples
Look in particular at my answers in the first 2 threads, I think you'll find all you need there. 
Good luck!
A: If I understand correctly, you're still an undergraduate student?  In that case, I would suggest either a study group or an independent (supervised) reading course.  I don't think you can learn to prove things by reading proofs over and over by yourself.  Find some people to work with, make sure there's an expert among you, and progress through the material together.  
A: I don't imagine you'll be able to have success studying higher-level material if you found you were unable to understand many proofs at this level.
I would suggest that you learn the same material again from a different, more accessible, source, and make sure you understand the proofs and are able to do problems. 
Elementary Analysis by Ross is an accessible analysis book for beginners. Spivak's Calculus is also excellent as a first introduction to proofs, and it comes with a solutions manual.
To save time, you can skip the chapters and/or exercises in which you're satisfied that you're already proficient in the material. 
