TITLE: What math is necessary to learn manifolds?
QUESTION [13 upvotes]: I want to learn about manifolds, but I'm only a senior in high school and obviously have a while to go.  I'm in AP Calc BC.  What should I study to eventually learn manifolds? Linear Algebra? What else?

REPLY [20 votes]: I hate to add yet another answer to the list, but I would like to be thorough.  The pre-requisites for learning about manifolds are as follows:

Multivariable calculus
Linear algebra
Real analysis
Point-set topology (a.k.a. General topology)

For multivariable calculus and linear algebra, most of the standard texts will do.  Note that multivariable calculus and linear algebra can be learned independently of one another (so it doesn't matter which one you learn first).
There are many good texts for learning real analysis, some of which are mentioned in the answers to this question.  It will be important that your real analysis education cover not only single-variable differentiation and integration, but also multivariable differentiation and integration as well.  In particular, it's important that you learn about the Inverse Function Theorem and Implicit Function Theorem.
Where point-set topology is concerned, some of it will hopefully be covered when you learn real analysis.  (Indeed, many analysis texts consider point-set topology to be a part of analysis.)  You will need to learn about metric spaces and topological spaces.  An excellent book for this is "Topology" (Munkres).
Finally, once you've gotten through all of this, I would say the text to use for manifold theory is "Introduction to Smooth Manifolds" (John Lee).  In fact, the book has an appendix at the end which gives a rapid treatment of each of these four subjects.  So, if you absolutely cannot contain your curiosity, this might be worth looking into.
Optional: One more helpful (but not necessary) pre-requisite is elementary differential geometry (or classical differential geometry), which is a beautiful topic to learn once you've finished with multivariable calculus.  My recommended textbook is "Elementary Differential Geometry" (Pressley).