How can a high schooler get more involved in mathematics?
For a high school student interested in majoring in math and learning more about math, what kinds of mathematical research can a student in high school get involved in? How can a high school student get involved? If a high school student does conduct research, what mathematical journals would be willing to publish their work? Also, what other ways are there for a high schooler to get involved in more complex mathematics?
From personal experience (either as a student or a teacher), here are my recommendations:
If you are near a university and can afford it, see if you can take math classes at the university in place of your high school math classes. Talk to the professors and see if they would be willing to have you in the class (they probably will, since they were probably in a similar situation at your age). This is how I got really into math, and it was probably the best decision I've ever made. If you can't afford it, you may still be able to audit classes, although I don't know if you could get out of high school classes for this.
I would recommend against trying to do original research in high school. I certainly tried and failed; it is only recently that I've gotten to the point where I can do original research on any mathematical problem, and even now I'm only really at that point for one specific problem. However, I would recommend reading the cutting edge research on some problem you find interesting. The first time you try to read a cutting-edge paper, you'll probably run into fifty terms you do not recognize. Looking up half of them will give you more terms you've never seen. It's a deep rabbit hole, but after a while you can probably develop the background necessary to understand some of the work on a current problem. I found that having a particular problem to focus on really motivated me to learn, and it eventually developed into original research!
Many universities have weekend and summer programs in mathematics. You can learn a lot at these, particularly if you are enthusiastic and make a point to interact with the staff, who are usually college math students. My personal experience is as a counselor for the YSP and VIGRE programs at the University of Chicago, which I would recommend if you are near Chicago, but I'm uncertain whether they will continue since the professor who directed them passed away a few days ago.
Pick one or two accessible textbooks, and try to read and understand them. Do all the exercises, or at least all the hard ones. My first textbook was Modern Algebra: an Introduction by John Durbin, which I think is quite good if a little on the easy side. Spivak and Baby Rudin are standard recommendations; by most accounts they are quite good but I can't speak on them personally since I've never read them. As more advanced books go, I rather like the Dover and Springer Undergraduate Texts in Mathematics series.
I would recommend visiting math.stackexchange.com, but that feels a tad redundant :)
Not to be a downer, but I've found that it's hard for even an undergraduate majoring in math to get involved in original mathematics research (from personal experience). The issue is that there's not a lot of low hanging fruit in mathematics that would lend itself to be a good project for a high schooler or undergrad mathematician. However, I don't mean to suggest you shouldn't try. Ask professors at local universities if they have any projects that might be accessible to a high school student or undergrad (perhaps provided you do a little background reading on your own to familiarize yourself with the concepts necessary). As an undergrad, I've been extremely lucky with "research type experiences": my freshman year, I approached my honors calculus professor about doing research. He let me down saying essentially what I told you, that it takes a lot of advanced background knowledge in mathematics to be able to do original research (simply because so much has already been done), but he also gave me the opportunity to try to solve an already solved problem that I didn't know the solution to (the problem was "how many ways can a positive integer $n$ be written as the sum of four squares?"). This project led me to study elliptic curves, modular forms, and other sorts of awesome advanced maths I never though I'd be doing as a freshman in college. So, even if you don't get involved in original research, you might still find an awesome experience.
On the other hand, I'm an undergraduate junior now and actually working on original research. You might get lucky and find someone who has a problem that's accessible with minimal background information, although you might have to work extremely hard (read: you'll most definitely need to work extremely hard) to catch up on that "minimal background information" (in my case, I needed to familiarize myself with a lot of category theoretic jargon and concepts before I could approach the problem at all). However, I am currently involved in actual research, and it's awesome. Don't give up! Even if you don't find anyone with an accessible project, you might still find people who have suggestions about what you can look at on your own. In high school, I didn't have access to anyone who was able to tell me about modern mathematics and modern research, so I found a subject that I was interested in (topology) and tried to learn it for myself. It's certainly not the same as actually doing research, but self study often presents many of the same problems to students as research: you don't know what to do, you might not know anyone who has experience working with the concepts you're struggling with, and you need to be very clever to overcome the obstacles you find in the problems (of course, the difference is that if you really want to, you can probably find the solutions online for something you're self learning, whereas the solutions most likely won't be easily available online when you face a problem in your research).
As for publishers, I'm not really sure what options there would be if you find research to get involved in. But that shouldn't be your goal anyway. You've got a lot of time; focus on finding something interesting and trying to understand it (even if it's been understood by others before) - that's a lot of what research is about anyway!
Edit: I see that the Ross and PROMYS programs are being mentioned in the comments. I am a little biased, but I believe that these programs are fantastic ways to be introduced to research style mathematics in an accessible manner as a high school student/undergraduate. Even if you aren't interested in number theory, you'll still encounter the same issues when facing research in any area - so struggling through number theory may provide valuable intuition/experience for struggling in the future in other areas of mathematics (it also helps that many deep ideas from number theory can be hinted at/worked on without introducing a lot of complex jargon). And while these programs are based on number theory, other aspects of mathematics become involved as well - so you wouldn't only be finding out about number theory, but you'd be learning about all sorts of awesome stuff (from analysis to category theory)! You should definitely look for things that interest you, but when it comes to a "research experience," these types of programs might be some of your best bets as a high school student. Don't dismiss them because you think you're not interested in number theory - there's a lot of valuable knowledge to be gained from struggling with any subject, and who knows? you might find you actually are interested in number theory after all!
In the end, one of the most valuable research experiences you can get is that of struggling with a challenging problem: there are different ways to get that experience, and not all of them involve actually doing research. Look around, keep your mind open, and ask questions - even if you don't wind up doing original research, you'll be well ahead of the game.
The best way for any talented high schooler to get involved in mathematics is just to learn as much of it as he or she possibly can. Doing research in it isn't realistic for high schoolers except for the occasional prodigy who's doing calculus by 12 years old. What you should do is see if an intensive program for high schoolers is available that allows a student to do a thesis before entering university-something like the VIGRE program at the University of Chicago.
As for any subjects you could attempt to produce original results in that don't require an extensive background, you could certainly try combinatorics. There's a reason why that's the traditional proving ground for students trying to begin research at a young age.
The standard book acting as a gateway for talented high schoolers into advanced mathematics has been Micheal Spivak's Calculus and it's hard to imagine better preparation then that.
Good luck!
First off, I am not a professional mathematician but I can share what got me interested in math and a life-long love of math (I am 60+). Mathematical puzzles, recreational math and elementary number theory. In my view, elementary number theory is the most fascinating field of mathematics because problems are easily stated and understood but are frustratingly difficult and challenging.
Good luck and wish you the very best.