Many students in the high schools of our world despise the hour or so that they spend in math class. I was one of them, and if I had to sit through more math classes in an American high school, I would most certainly be one of them. Many adults hated math in their youth, but now have a different outlook and want to learn about it. If you are someone who wants more from mathematics, and want to see what it really is, then you must begin the long journey toward understanding what Carl Gauss deemed "the queen of the sciences."
I would suggest at first that you learn calculus and linear algebra. In order to learn these, you need Euclidean geometry, high school algebra, and some trigonometry. All of these can be learned in depth on Khan Academy for those who don't know them. Getting back on track, calculus is the study of very small changes and behavior of functions at infinity. I learned calculus using The Great Courses Plus' courses on calculus (Make sure you do calculus 1, calculus 2, and multivariable calculus) as well as using Khan Academy (In which case do AP calculus BC and multivariable calculus), the latter is completely free! Now, in order to understand calculus, you should understand some trigonometry, geometry, and algebra. Calculus is the first step into a beautiful and deep field of math called analysis, for which I will devote a paragraph below. Now, linear algebra is a subject which many people enjoy, and the crucial object of study in linear algebra is an object called a vector space, the elements within it, and special functions between vector spaces called linear transformations. Linear transformations are represented by something that you are likely familiar with, matrices! The best place to learn linear algebra is likely MIT Opencourseware, where the linear algebra course is taught by one of the great teachers of our time, Gilbert Strang. Alternatively, Khan Academy has a short course on linear algebra, but I recommend the MIT course, if you want a more in depth understanding.
After learning both of the fascinating subjects of linear algebra and calculus, I hope that you are ready for more, and not burned out. Both topics can be difficult for students, but if you truly want to learn mathematics, you must press on!
Moving on from calculus and linear algebra, it is time to get ready for deeper mathematics. To do this, set theory and a knowledge of proof techniques are a must. I recommend learning how to do proofs before learning set theory. Some good books are How to Solve it by George Pólya and Book of Proof by Richard Hammack (This one is free!), picking up either of these books and doing the exercises within will have you ready to do real mathematics in no time!
Now we get to the good stuff! Specifically, topology. Topology is one of my favorite fields of math (along with geometry, number theory, & mathematical physics) and it is one of the most widely used. You will find that topology is used almost everywhere in mathematics: it is prevalent in analysis, geometry (topology is often called 'rubber sheet geometry'), number theory, representation theory, combinatorics, and the list goes on. Topology is the study of certain mathematical spaces called topological spaces and the properties that are preserved among 'equivalent' spaces. The notion of equivalence in topology is different from what you might think, two topological spaces might look completely different from a local point of view, but they can be 'equivalent', since the study of topology is not concerned with local properties. To learn topology, I recommend Introduction to Topology by Bert Mendelson (a book that I am currently using to study topology, along with other resources) and Topology Without Tears by Sidney Morris, the first book is under 20 American dollars and the second is free (the pdf is linked). Topology combines with the next subject, abstract algebra, to form a difficult, but important field known as algebraic topology.
The reason that I like abstract algebra is that the problems are fun and interesting. When I first started to learn it, the introductory exercises felt like puzzles to me, and the more advanced exercises are challenging and profound, especially in the awesome book that I am about to recommend. Abstract algebra is the study of algebraic structures, which are like sets, but equipped with operations like addition and multiplication (more generally, the operations are arbitrary). It is probably more far reaching than topology, because virtually every field of math deals with some type of algebraic structure. In a first learning, I recommend Charles Pinter's A Book of Abstract Algebra, It is a cheap and invaluable resource for learning abstract algebra, and the discussion is very informal, which makes it an easy read.
Finally, we reach analysis. The subjects of real and complex analysis are like night and day. Complex analysis is beautiful and wonderful, but the subject of real analysis is difficult and sometimes uninteresting. However, if nothing else, one should learn real analysis in order to apply it to something more advanced like analytic number theory or functional analysis. This rambling might not make since yet, because I haven't even told you what analysis is. Analysis is the study of calculus over different spaces. Real analysis works over real numbers, complex analysis over the complex numbers, functional analysis works over infinite dimensional vector spaces (I'll admit, this sounds kind of cool, but you must learn real and complex analysis as well as linear algebra to work comfortably with it), etc. The crucial part here is that calculus is a must in order to study analysis, not so much for real analysis than for complex, since real analysis is essentially where we prove results from calculus. However, I wouldn't learn real analysis without calculus first. I recommend Understanding Analysis by Stephen Abbott for real analysis and Visual Complex Analysis by Tristan Needham for complex analysis.
This might not all be accomplishable during high school. If you are homeschooled, you can probably do it, but if not, you can learn these subjects in your free time and during holidays. If you are a functioning adult, I suggest learning it in your free time, on the weekends, etc. This is just a guide to learning pure mathematics on your own, especially for impatient persons like myself, who can't wait until college to learn these beautiful ideas. If you don't get the chance to learn all of this before college, learn it during college, if you have time. This is not all there is to pure mathematics, but it is a solid foundation for those who want to continue their study of math into fields like differential geometry, algebraic geometry, symplectic topology, functional analysis, several complex variables, number theory, dynamical systems, and mathematical physics (you might want some knowledge of physics for this one) just to name a few. I wish you luck on your mathematical journey, it will be hard, but only after overcoming the rough terrain of the mountain can one see the beautiful view from its peak.