Is there a good book that combines Calculus I, II, III, Analysis I, II with the routine calculation exercises and that also shows rigorous proofs of the theorems? I have looked at Vladimir Zorich's Mathematical Analysis I but does not have many routine questions in it. So what are your suggestions? So the idea is I want to study Calculus with Analysis together.
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Check out Spivak's Calculus book.
From the preface:
"Every aspect of this book was influenced by the desire to present calculus not merely as a prelude to but as the first real encounter with mathematics. Since the foundations of analysis provided the arena in which modern modes of mathematical thinking developed, calculus ought to be the place in which to expect rather than avoid, the strengthening of insight with logic. In addition to developing the students' intuition about the beautiful concepts of analysis, it is surely equally important to persuade them that precision and rigor are neither deterrents to intuition, nor ends in themselves, but the natural medium in which to formulate and think about mathematical questions."
Here's an MSE post about it: Spivak's Calculus?
And then for multivariable, you might look at his Calculus on Manifolds. But that may require more background -- it's been a while since I looked at it.