Self-Teaching: Is Geometry the Nexus of all Mathematics? Necessary prologue: I'd really like to become more fluent in the language of mathematics. I don't have a schedule that permits me taking a class and any on-line tutors that I find seem relatively sketchy. So, I'm looking to teach myself. 
After reading quite a few books on the history of mathematics, it's become clear to me that geometry seems to be where mathematics all started. Not only that, but it seems to lead into other branches of mathematics quite well (for example, everything geometric can be discussed and represented algebraically, so I'll, by necessity, bump into algebra along the way). 
My question: Given that I'm looking to teach myself, that I have no direct NEED for understanding a specific branch of mathematics, and that I intend to be studying casually, like, for the rest of my life off and on, is it fair to assume that geometry is a good starting place for a general mathematics education?
 A: First: I love this project. All the best to you. And I think you're going to have a great time.
Second: Mathematics is too big for this question to be able to be answered quite in the way you mean it. It is so big that not even a life of devoted study could lead any individual to an understanding of more than a small fraction of what's already well-understood by humanity as a whole. It is often said that while it used to be possible for the world's greatest mathematicians to have a birds-eye view on the subject as a whole, the last generation of mathematicians to have such a view was the generation of Poincare and Hilbert, 100 years ago. (Since you've read a lot of math history, you've probably heard this idea before.) The subject has sprouted in so many directions it's even difficult to keep track of them all. Ask a researcher in dynamical systems what the most important or fundamental math to know is and you'll get a totally different answer than if you ask a number theorist. So your choice of where to start is necessarily going to be a little bit arbitrary in relation to the question of "most central."  What counts as the center depends on what part of the boundary you're looking at.
One piece of good news is that to quote math writers Bob and Ellen Kaplan, "anything leads to everything." Just as you're going to bump into algebra if you pick up geometry, you're going to bump into geometry if you pick up algebra. Basically you're going to bump into both if you start anywhere.
That said, I don't want to ignore your request for concrete advice about where to start. This request actually goes beyond the question of what field to start with. Does "geometry" mean Euclid? Or a high school geometry textbook? Or a book on projective or hyperbolic geometry, or the geometry of Einstein's theory of relativity? So let me suggest some actual books you might try. The three books I'm about to suggest have the advantages that they are a) aimed at educated laypeople, and b) broad in scope. (In fact, they each attempt to take a wide-angle view of mathematics as a whole. Of course a propos of the last paragraph, math is too big for this to be done in an objective way: that's why the 3 books are extremely different from each other.)
Here are the books:
What Is Mathematics? by Courant and Robbins
The Heart of Mathematics by Burger and Starbird
Mathematics and the Imagination by Kasner and Newman
The first and third are bona fide classics; the second hasn't been around more than a few years but it is wonderful.
All 3 books contain real, significant mathematical content, but What is Mathematics? and The Heart of Mathematics have the advantage that they contain exercises and problems, i.e. will actually get you doing math, not just reading about it, so they may be the better choice. My recommendation is you have a look at these two books online or in the bookstore, and buy one of them, and start working out of it.
A: Given that you have the freedom to choose your own path, you might as well study what you find interesting.  Geometry is certainly a good place to start, if that interests you.    
