Need Suggestions for beginner who is in transition period from computational calculus to rigorous proofy Analysis

Apostol's Mathematical Analysis for a beginner? No way! You'd be bored to death presumably, and also you'd be scared, which is not at all good if you are beginning the subject.

Among these three, I would go for Tao's Real Analysis. The reason is simple: I don't want to be intimidated by a subject I am about to learn. A lot of mathematical topics need good exposition, and Terence Tao is a brilliant writer, it seems.

G.H.Hardy's book is good, but old-fashioned.

See, the point is, you need a book that will act as a mentor, that will firmly clasp your hand and guide you through the depths of the subject. I was fortunate enough to get a brilliant professor who made Analysis a cakewalk, which is really not so simple to teach.

Hence I would suggest a combination of Bartle-Sherbert and Tao if you want to begin understanding what Analysis is all about. You can enter the advanced topics presented in these books as well.

And once you are confident enough, you can go through Apostol or Rudin, who are invariantly the masters. But, it's like learning magic: you need to first learn basic, easy tricks before you can turn into a good magician. Keeping this in mind, if you follow what I said, things will be easier.

A lot of undergrad math is pure rigour, which produces a sense of completeness and satisfaction after a proof, but can seem too abstract or useless to someone who has not undertaken a thorough course. Calculus courses help in computation of integrals, finding clever substitutions, etc. but in real life, it rarely matters now. You'd try your hand at a hard Integral computation only if you have nothing else to do; otherwise, talking practically, you'd just look up Wolfram Alpha.

All the best.

In my opinion, the best introductory book to real analysis is Berberians's A First Course in Real Analysis.

The content in the book is much lower than it is in Rudin's book.

I see that you have been suggested Tao's book by Landon Carter. I do not mean to undermine his suggestion in anyway. Tao's book, if I recall correctly, defers most of the proofs to the exercises. For someone who has never seen rigorous proofs before, reading such a book might be a bad idea, especially if you do not have a friend who you can readily go to whenever you get stuck.

Berberian's book is a really gentle introduction to the subject and in my opinion very well written. Just check that book out and decide.

As for starting Modern Algebra simultaneously, it would be a good idea to start group theory from Herstein's Topics in Algebra.