Am I too young to learn more advanced math and get a teacher?
I was recently in a similar situation. After finishing precalculus at my high school, when I was 15 I started taking calculus at my local university and studying higher mathematics on my own (out of the book "Modern Algebra: An Introduction" by John Durbin, which in retrospect seems laughably basic but at the time blew my mind). Three years later, I can say without a doubt that it is the best decision I've ever made. I ended up learning mathematics through a combination of taking classes at university, talking with students/professors, reading textbooks, and using this site. I did have one major advantage over you though, as my parents are both professors (although neither of them math professors) which made it easier for me to get into classes. However, I know of other people doing the same thing without any connection to the university. Here are some things I would recommend based on my experience:
Get an introductory textbook for some relatively advanced subject, such as Calculus, Linear Algebra, or Abstract Algebra. Read reviews online before choosing one to find one that is both rigorous and easy enough for beginners. I'd recommend Spivak for Calculus (take this with a grain of salt though, as I never read it but have heard good things about it) or Durbin for Abstract Algebra. Make sure it comes with plenty of exercises, and DO THEM. If you don't know how to do a problem, or if you've done it correctly, ask here!
If you have a university nearby, take advantage of it. Email a professor teaching an upcoming introductory course and explain your situation to him/her, and ask if you can sit in on the class. You might even be able to enroll in classes as a non-degree-seeking student, if the university allows this (most do) and you can afford it (if it's a state school, the tuition for a single course might not be too bad). Don't be afraid to talk about math with professors. It can be intimidating, but remember, these people have dedicated their lives to math. Almost uniformly, they LOVE it. Half of the time I had to find a way to break off a conversation with a professor because they were so engrossed in the math at hand.
Find something specific you don't understand. It may be a theorem, a proof, a concept, or even an unsolved problem, so long as it fascinates you. Figure out what you need to know in order to understand it, and start down the rabbit hole. The experience of coming to understand something like this can be very rewarding in addition to teaching you a great deal of mathematics. I've had several of these over the past few years, most recently an unsolved problem known as the Triangular Billiards Conjecture which I'm studying right now.
If you have any questions about my experience, feel free to ask. Good luck!
There is no reason to delay looking at whatever you want to look at. For that matter, I would say that you have no obligation to "systematically" read anything, or do exercises, unless you want to. In the U.S., not only is the high school math curriculum stultifying and anachronistic, but also most of the undergrad curriculum is the same. In particular, I'd recommend not being toooo trusting of "standard textbooks", because the design of conventionally-published textbooks is very often strongly influenced by pressures from publishers to make the things match the usual curricular structure.
Looking around either in physical libraries or on-line stuff at _non_textbook_ writing (at the very least as a supplement to textbooks) gives a much better perspective on "where math is going", and ideas, rather than "lists of required topics". Further, I think it is misguided to tooo strongly require "mastery" of a given thing before looking at "the next thing"... because the purpose of a thing/idea is often only revealed by what happens later, and that hindsight often makes many of the earlier details much more intelligible. It is important to allow oneself the flexibility to "skip forward to see what happens", all the more so given the tendency/tradition of mathematics writers to logically re-order things: instead of giving illuminating examples with observable properties, subsequently organized by choosing terminology, the style is to give definitions first, etc.
Also, with textbooks, beware that many of the exercises will be simply "made up" to meet expectations. In particular, they may not really be worthwhile, but just filler. Hard to know which is which, for a beginner, too. In extreme cases, "doing exercises" can actually be a strange avoidance mechanism that obstructs learning! :)
Also, the contest-math world is not very much about mathematics, although if you have a flair for this it can be a good PR vehicle for you, and a way to meet other people approximately interested in mathematics. The down-side is the obvious emphasis on speed and cleverness, rather than learning serious mathematics, so keep in mind that this is not at all "mathematics proper".
Again: trust your own interests, judgement, curiosity.
Here is a link to the Berkeley Math Circle:
http://mathcircle.berkeley.edu/
It is replete with really outstanding teachers and offers lots of material for study at many levels. Maybe you're lucky enough to live near there.
Otherwise, if you go to the "Links" section, there is a further link to other math circles.
I would venture that through their generous endeavor you will find something.