Is the multivariable calculus in Apostol's 2nd Volume enough for Spivak's Differential Geometry Series? The question is basically in the title. Will I need something more along the lines of Munkres' Analysis on Manifolds or Spivak's earlier Calculus on Manifolds, or will Apostol suffice? 
 A: The prerequisites for a course I took in Spring 1980 (at a U.S. university) that used Spivak’s Volume I were the following three courses:
1. Graduate linear algebra, which roughly covered chapters III, XIII, XIV, XVI in Lang’s Algebra, but I think knowing the first 5 chapters of Hoffman/Kunze’s Linear Algebra would easily be sufficient for the course I took, and the only reason for requiring the graduate linear algebra course (rather than our advanced undergraduate course that used Hoffman/Kunze) was for the mathematical maturity obtained in that semester of my university’s graduate algebra sequence.
2. Graduate point-set topology, which used Topology. A First Course by Munkres, covering most of the book except some of the specialized stuff on nets/filters, metrization theorems, etc. in the middle parts of the book, and all of the algebraic topology material plus most of Massey’s Algebraic Topology: An Introduction, although the material in Massey wasn't needed for the course I took. (I believe one reason for the rather extensive treatment of algebraic topology that was given in that one semester first graduate topology course is that most of the graduate students had previously taken an advanced undergraduate course in point-set topology, something that was probably more common in the U.S. in the 1970s than is the case now.)
3. Advanced undergraduate level differential geometry, which used either Do Carmo’s Differential Geometry of Curves and Surfaces or O’Neill’s Elementary Differential Geometry (depending on who taught the course).
Lebesgue integration and measure theory was not required (although I suspect that everyone who ever took this course had previously or simultaneously taken a measure theory course), but an upper level 1-year of advanced calculus or upper level of at least one semester of something that includes the multivariable parts of Rudin’s Principles of Mathematical Analysis is implicit in the above (being required for the topology and differential geometry courses).
However, the above were the preprequisites for the specific course that I took. Depending on who taught the course, the text that was used was either Spivak's Volume I or Warner's Foundations of Differentiable Manifolds and Lie Groups. Regarding what I think would be prerequisites specifically for studying Spivak’s Volume I, I think Spivak’s Calculus on Manifolds and some basic knowledge of metric spaces is minimally sufficient. For someone with a weak background in multivariable calculus, perhaps better than Calculus on Manifolds would be Advanced Calculus of Several Variables by Edwards or Functions of Several Variables by Fleming.
My recommendation is that you begin by going through Spivak’s Calculus on Manifolds (and learn some minimal metric space concepts if you don't already know this), and then begin Spivak's Volume I, having on hand (for reference and for collateral reading, as needed) some texts like the above and Analysis on Manifolds by Munkres (which definitely falls under the collateral reading category and not under the prerequisites category).

(from p. vii of Volume I of Spivak’s 1979 2nd edition) These notes [this almost certainly applies only to Volume I] were written while I was teaching a year course in differential geometry at Brandeis University, during the academic year 1969-70. The course was taken by six juniors and seniors, and audited by a few graduate students. Most of them were familiar with the material in Calculus on Manifolds, which is essentially regarded as a prerequisite. More precisely, the complete prerequisites are advanced calculus using linear algebra and a basic knowledge of metric spaces. An acquaintance with topological spaces is even better, since it allows one to avoid the technical troubles which are sometimes relegated to the Problems, but I tried hard to make everything work without it.

A: The main components that you need to be comfortable with to start studying differential geometry -- other than multivariable calculus and linear algebra that you can find in Apostol's II -- are basics from point-set topology and multivariate analysis/metric spaces.
Respectively, you need to have a firm grasp on
1) Abstract topological spaces, separability (e.g. Hausdorff), compactness, connectedness, continuity, homeomorphisms. Note that you will likely have heard of these terms before but maybe not in the context of abstract topological spaces which generalise them.
2) Proofs involving multivariable functions, metric spaces (which are more abstract than $\mathbb R^n$ and thus prepare you to the study of manifolds), multivariable Taylor expansion.
If I were you, I would start working with Spivak's differential geometry series, while having a copy of the more elementary books which you mentioned plus a book on point-set topology, to refer to them when you have a gap in knowledge. This might be tedious to start with, but you will save time compared to reading all those books beforehand. This is assuming you can borrow them from a library as opposed to buying them. Otherwise, you are probably better off starting with Spivak's earlier Calculus on Manifolds.
