(Soft) What maths should I concentrate on at 16-18 years old? Some background information first of all:
I'm 16 now and just started studying mathematics intensely. I live in the UK and my goal is to eventually become very good at advanced mathematics (graduate level and up) but that is still far in the future. I'm about a year or slightly more ahead from the expected curriculum knowledge in my age group.
The advice I'm seeking of is that I am not sure whether to keep going ahead of the curriculum and eventually learning university maths a year before actually going to uni or should I try to improve my problem solving skills by for example trying out Mathematical Olympiad questions. If I do MO instead then I fear I might be wasting time (even if I'm improving at problem solving) when all I really want to do is the more advanced stuff which is more proof laden I hear anyway.
To answer the question it might be helpful to know what my intentions for pursuing mathematics are. I enjoy learning new fields of mathematics that I did not know of before. A new topic excites me, a new problem to solve with the same tools just doesn't have the same effect on me mentally. I can pick up stuff quiet quickly and get bored if I'm forced to keep doing what I already know without any new knowledge. I realise however that mathematical ability is not just about knowledge but also problem solving skill.
 A: Basically, if you're just interested in what topics to learn next, you really could start on almost anything.  Your choices roughly come down to 


*

*foundational math (set theory, logic, proofs)

*geometry

*linear algebra

*calculus/ real analysis (at the level of Spivak)

*abstract algebra

*number theory

*probability theory


Each has their benefits.  I'd suggest taking some time to research each one and trying to figure out which grabs your attention the most.  Then you can invest a more substantial amount of time and figure out if it really is someone you enjoy.  If not, you've definitely got time to change your mind.  The point though is to just find something you like and run with it.
A: It would be  nice if you could learn some basics of calculus.
But be sure first that you are already good at the very basics of algebra, identities, logarithms etc.  
Do not try Math Olympiad problems.
In my opinion, they train you to be a good "warrior" for 3 hours (the exam) but a mathematician is not a warrior and certainly has plenty of time. 
Try to understand the proofs you read and not only memorize the theorems. 
Along with Calculus a good advice would be to study Number Theory.
This way you will see what mathematical genius actually is. Trust me!
A: I would recommend reading Velleman's How to Prove it, which teaches you about how to think logically and construct mathematical proofs. This is often the biggest hurdle for new mathematics students: figuring out what constitutes a proof, what are logically valid arguments, and how to present a proof. 
Alongside Velleman, I would recommend Axler's Linear Algebra Done Right. Linear algebra is the study of vector spaces (and later on, modules) and the linear transformations between them. Linear algebra plays a key role in every field of mathematics, as well as in physics and computer science. For example, the derivative operation in calculus is all about linearizing functions. 
A: Nothing is wasted. Hard problems which force you to think are always good. Cambridge University has a reading list and there is more information on the website. And I guess other faculties will do something similar.
A: My personal experience in the US is that there is a vast difference between the typical high school curriculum (algebra, calculus which are primarily learning methods) and what might be considered pure math. 
The ability to prove things (as several others have mentioned) is paramount. This also entails recognizing what conditions are necessary and sufficient, where thins can fall apart, and a wonderful exercise in self-reflection to consider if you really know what you are doing. 
That said, I feel (once you have a familiarity with calculus, also mentioned above) that you persue a particular course. Many programs begin with real analysis. The benefit of this is that math knowlegde is often cumulative, and thus following a course will have the added benefit of seeing how and where prior developed material comes into play. 
This is quite different from a non-math or science program, as you can read any Shakespearian play or Wordsworth poem without any prior experience. 
If you are interested, here is a link to a free down-load of a RA course given by Vaughan Jones (Fields Medal). They are self-contained meaning they start at the beginning, and you need not look elsewhere as you progress. (Notice however the first few pages are a bit technical for thoroughness, so take a glance, and start a page 4.) There is nothing like the real thing.  
https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxtYXRoMTA0c3AyMDExfGd4OjJiNTJkM2M2ZWUzZGIwYWQ
A: I don't know, where you are from (EDIT: okay... I do, nevermind). This kind of matters, because education differs between different countries (at least I know, here in Germany it is different from math at American universities). The most difficult step is going from the "pre-rigorous" to the "rigorous" state. In Germany this is more important, because rigor starts with the first year in university. I believe it is different in America, where one learns "Calculus" before "Real Analysis" and "Linear Algebra" means matrix arithmetic, not "theory of finite-dimensional vector spaces".  In Germany for example, there is no "calculus", we start with "real analysis" immediately. However, at some point or another you have to make the transition from understanding things only by intuition to proving things. The best advice I can give you:
Practice, practice, practice!
More specifically, I personally benefited by learning some of this stuff first:


*

*propositional logic, meaning of quantifiers 

*(naive) set theory, including membership, subsets, union, intersection, Cartesian products, relations, orderings, equivalence relations and quotient sets, functions, tupels, 

*mathematical induction


...and from here you can go anywhere! Well, starting with abstract algebra (groups or vector spaces) is a good idea, in my opinion. If you prefer it, do analysis (just know, that proofs in analysis tend to be "messy and complicated")
A: I was in a somewhat similar situation('discovered' math in mid-end of HS, thinking about going into competitions or directly to uni level math) and I choose the latter option. I would suggest that before doing anything else you should get your hands in an "advanced" calculus text to learn what a proof should look like(e.g. Apostol's or Spivak's for a gentler introduction) and learn some logic/set theory with something like Halmo's Naive Set Theory since both are used everywhere.
A: TL;DR 
At this stage all you need to care about is to keep stretching your brain in many different directions. There still will be a lot of time for you to learn all the amazing theories, useful tools or amusing curiosities. Let your imagination run free and enjoy the math you do (but be serious, don't slack off).
If I were to give you one advice on what to do: experiment!

Disclaimer
Each mathematician has his/her own best set of things to do and these differ considerably. I have no idea what will work best for you, I can only suspect and even that is warped by my own subjective perspective. Of course, I have some observations about what worked and what did not for my friends or students I taught, on this observations and various musings of others I based the advice that follows. However, you won't really know what works until you try it - each to his own.
General advice
If you had to choose between knowledge and thinking, at this stage I urge you to pick thinking.
Both are important, but there is only a finite amount of time you can spend on them and


*

*you will pick up some knowledge along the way and that will be enough to prepare yourself for more knowledge in the future,

*you will be able to learn new things for many, many, many years,

*there are some quite good substitutes for knowledge (e.g. books, internet),

*I know no conclusive research on this topic, but to me it seems that thinking capabilities work similarly to other human traits, that is, the early years determine the limits you can later achieve (the cap on your abilities).


It doesn't really matter that much what kind of math do you do, as long as you keep stretching your brain in different directions. Also, whatever you do, be serious, don't slack off.
More personal advice
Personally, I would advise you against learning advanced material and study university topics ahead of time (more on this below). Instead, practice a lot of Math Olympiad problems, but optimize for brain-stretching, not for contests (more on this also below).
Another thing that would be great to do is to experiment with math. Take some math and use it, or find some practical (getting purely abstract too soon won't do you any good) problem (it might small and insignificant, doesn't matter) you would like to solve and try to do it with math (perhaps learn some new things in the process). 
I will not only give you an example, but three tasks that will be great for your thinking (they will take some time, but not that much):


*

*Make a simulation of a lunar lander (the graphics can be trash) according to physics equations, including (at least) mass, gravity, atmosphere density (to have some damping effects) and thrust forces from engines. Represent them as differential equations and solve numerically (use something simple like Verlet).

*Transform an audio voice recording (e.g. some MP3) using your own FFT algorithm (it can be DFT, doesn't matter), e.g. to be a bit higher. (To convert the numeric data to and from audio files use some tool or library.)

*Code a minimal raytracer. Try to use homogeneous coordinates inside and Monte Carlo methods during rendering. Also, play with transformation matrices.


If these sound intimidating, don't worry, they are much easier than they seem and you can find a lot of help online (tutorials, forums, stack-exchange sites, etc.). Besides, you don't need to understand all the details of the underlying theories, to complete these tasks you only need to learn a few bits for each (e.g. derivatives and integrals, complex numbers and polynomial evaluation, matrix multiplication and some randomization). In fact, a few paragraphs below I will argue that you should not learn too much advanced material too early.
The above tasks require some programming skills, but I guarantee that coding ability will be useful to you (see also here). Each has an underlying topic (differential equations, FFT, linear algebra) which is highly relevant, frequently used in applied math and amazing by itself. You will have a better understanding of what math can do, and believe me, doing it personally is a whole other experience than just hearing about it. Finally, it is great fun seeing it run!
(If you are more focused on abstract math, these problems will still be helpful.)
On rediscovering the wheel
Whatever you will ultimately do, at your age, don't be afraid of rediscovering the wheel. Although in research it's better to rely on known theorems rather than prove everything from scratch (human lifespan is limited), there is no better way of learning how to invent new things (I would recommend against true open-ended research, because it's easy to get frustrated).
Basically all standard textbook problems rediscover the wheel (they were solved before, weren't they?), but somehow we understand that this is precisely their intention. However, ordinary theorems or folklore facts can also be though of as challenges. This is standard in geometry, so why not in other areas? For example, could you solve the quadratic equation without knowing/using the exact formula for roots?
On university curriculum
I have nothing but anecdotal data to support this claim, but I think that studying advanced books too early may not be such a good idea. You can find a more detailed description of my worries here: Effects of early study of advanced books, but to briefly summarize main points:


*

*such approach favors knowledge over thinking,

*when you study on your own, you might miss some important pieces,

*you train yourself in following the beaten track (rather than creating your own).


On Math Olympiads
Math Olympiad problems are one of the best mind stretchers I know. If you approach them right, a lot of them (depends on country) become great tryouts for your out-of-the-box thinking.
The most important thing here is not to optimize to get far in the contests, but to expand your mind. For example, there is a certain set of mathematical theorems which with enough practice allow you to solve many olympiad problems with little or no insight (e.g. the Muirhead's inequality is notorious for killing fun). So when you spot a problem that is easily solvable by some "overpowered" theorem, either try to find another solution or go on to next problem. Similarly, do not measure your time when solving Math Olympiad problems. If you are stuck, try harder (rather than moving on or reading solution). If you can't stand the problem anymore, leave it and perhaps try again next week.
Conclusion
Don't just learn math, or as Paul R. Halmos have said: Don't just read it; fight it! (the picture comes from the Abstruse Goose webcomic).

I hope this helps $\ddot\smile$
