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.
