I have done Chartrand's "Mathematical Proofs A Transition to Advanced Mathematics" (Relevant Sections) and am about to finish Abbot's Understanding Analysis.
So, should I go for Spivak's Calculus after Abbot's Understanding Analysis or Pugh's Real Analysis?
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I went into Spivak's Calculus with highschool knowledge and did just fine. Manage to solve almost every problem and grasp all the concepts; you should be more tha prepared for it. In fact, you might be over prepared for it, consider a more advanced text. Perhaps the first half of Calculus on Manifolds (again, by Spivak) may be useful in your case. The first half is a pretty good for multivariate Calculus and could serve as a first step towards general topology which is based (I am very biased).
Update: I see you mentioned functional analysis and measure theory. All the more reason to choose a book aiming for general topology as a next step.
If you have read Abbot's Understanding Analysis you have also covered almost everything in Spivak's Calculus. I suggest you to move to books covering further topics. I suggest: Mathematical Analysis by Tom Apostol.
It covers everything that is in Spivak's Calculus and further topics like Lebesgue integrals, Fourier series, multivariable calculus, multiple integrals...