I want to learn mathematics to extend myself. I am a middle school student that is highly fascinated by mathematics and its elegance. However, I have found that to appreciate a lot of mathematics a lot of knowledge is required. 
Currently, I can do basic arithmetic, basic geometry (angle sums, similar triangles, finding an angle), permutations and combinations, solve basic algebraic equations, simplify expressions, solve simultaneous equations, index laws, multiply and add matrices and arithmetic and geometric sequences.
So, I want to learn trigonometry, quadratics, logarithms, parabolas.., complex numbers and if possible a bit of calculus.
My goal is to have a good understanding and be able to apply the majority of these subjects within 4 months. Additionally, I am willing to work as hard as is needed to pursue these subjects.
All I would like is a bit of guidance of where to begin and whether there are any good resources that would be useful.
Help would be greatly appreciated.
Thank you so much :)  
 A: I highly suggest that you begin with learning elementary number theory.
You will understand different proof techniques
Number theory is kinda like a puzzle, It will enhance your problem solving skills and will you will also learn a lot of different notations that is used in all over mathematics.
Unlike complex analysis or calculus, I find number theory to be very exciting and encouraging for students who want to learn maths at a young age like you. That's why you will find a section in the mathematical Olympiad challenge about number theory.
There is not a lot of videos that I found interesting on number theory on the internet.
However, I highly suggest to read this friendly text book
Introducation to mathematical thinking
It is very easy to read and it's one of very few maths books that you won't actually get bored easily reading it
A: Thanks for your interest! It's always nice to see such dedicated young individuals. Since you live in the US, I feel obligated to say that as a former student, the US's education system is pretty screwed up; learning here is based on age rather than intelligence. What you want to do is perfectly normal, as the math system is going too slowly for you, but I suggest that you or your parents keep negotiating and pushing the school to accelerate you in your mathematics courses. Even just one year ahead can be a great benefit. It took us forever to allow me to be able to get the school to recognize the geometry I was learning on my own, so I definitely know the feeling you have.
As far as resources go, you should try borrowing an algebra II/trig textbook from the high school to learn the basic materials of trigonometry and algebra (conics, logarithms) that anyone should know. From here, you could consider learning calculus, but it may be better for you and the school to wait until you actually have a class in the course. Instead, expand your mind in the other numerous and accessible fields of math at this point in your life. You should look at more number theory, (more) geometry, combinatorics, and even linear algebra, as these items are not traditionally taught in a US school (save basic geometry). If you're looking for books on these subjects, I recommend Dover Publishing (and any introduction/beginner books they have) as they are relatively cheap and of good quality.
Additionally, you could look at recreational mathematics, which are the maths of novelties and puzzles. These are meant purely to enjoy and have fun with. Again, if you are looking for books, look at Matt Parker's Things to Make and Do in the Fourth Dimension or any logic books by Smullyan. There are of course, countless others.
The worst thing you could do at this point though is stop. Keep pestering your school to move ahead and keep looking around the internet for things to learn. Best of luck.
A: Textbooks
The clearest way to learn maths is to get your hands on a good textbook. I find that reading a textbook is the best way to study any subject.
When I was in high school (grade 8 to 12 in South Africa), I started reading through a calculus textbook which had been lying around in my house. A calculus textbook will often summarize the key concepts of trigonometry, logarithms, functions and polynomials in the first few chapters, and goes on to discuss topics that would be taught in your later high school years and first few years at university such as limits, differentiation and integration. A calculus textbook will extend you however, and you may want to start off with a textbook written for the higher grades of school.
Another option, which is more geared to the pure side of the science, is to read a book about logic and set theory. That will accustom you to the notation used in proofs and in defining mathematical structures.
Online tuition
If you have a specific question, searching online can often give you useful papers to read through. Also, there are websites geared for tuition such as https://www.khanacademy.org/ .
EDIT: I have just clicked on an advert for the MIT courseware: http://ocw.mit.edu/courses/#mathematics and this looks very promising!
A: I would also recommend looking into communities of other people who are also interested in math! There are several advanced summer programs you might find interesting, such as Canada/USA Mathcamp. In addition, there might be a math circle in your area [link omitted b/c of rep]. You could also look online at sites like The Art of Problem Solving.
At least for me, the community of other people interested in math was as important as (or more than) the specific math that I was learning. All of the topics you've listed are great topics to study, but I'd encourage you to look beyond the standard curriculum a bit, because there are a great many beautiful, satisfying, and approachable results that never get mentioned in the traditional school curriculum. Number theory (as mentioned by another poster) is a particularly fruitful area of study in that regard, but there are many others. (I've sadly drifted away from pure math in recent years, so I'm not the best person to suggest others.)
A: When I was your age, I thought exactly the same thing. I decided to study calculus because that was the "most complicated" mathematics I knew of. I found a copy of "Calculus" by Harley Flanders, Robert Korfhage and Justin Price (1970). I kind of lucked out, because even today, this is considered one of the easiest to understand and most comprehensive books on calculus ever written. At the age of 13, I spent the entire summer working through it. Needless to say, I crushed high school pre-calc and calculus, when I studied them 2-3 years later. It also saved me time, because later on I could spend less time on math and more on other subjects. Ultimately, this effort was a factor in helping me get into Princeton, where I went to college. Princeton is the top mathematical school in the world.
Nevertheless, I would not do this if I had to do it over again. When I got older I found out that you learn a lot more if you study what you are interested in, not if you follow a textbook. What you should do is study problems that fascinate you, then only use the textbooks to help you solve those problems. Use your own intuition as your guide. Ramanujan, possibly the greatest mathematician in history, learned mathematics by gathering discarded kraft paper from the docks in Madras and writing on them in charcoal. He made up and answered his own questions--and that is the way to do it.
I would recommend starting with the theory of numbers, rather than calculus. There is a book called "The History of the Theory of Numbers" by Dickson, which you can get for free from Google Books. Learning modular arithmetic is probably the number one best mathematical skill you can acquire to start with.
For learning calculus what I would recommend is starting at the end, not the beginning, the end being partial differential equations (PDEs). PDEs are by far the most useful thing in calculus, allowing you to calculate any rate equation. This is enormously useful in physics and chemistry. Unfortunately, in a normal calculus curriculum you do not learn PDEs until the very end, often not until studying advanced calculus in college. I would advise starting with PDEs and going backwards from there. Whenever you get stuck on a PDE problem, work backwards to learn the foundational skills you need to know to solve just that one problem. Once you can solve any PDE you will have a really valuable skill.
