Do most major math journals not care about a piece of research unless it proves an established result or problem? And by established, I mean something that of course already exists be it in a textbook or another research text. I ask because I am developing/researching a what could be said to be quite original concept on my personal time but I feel that it is too idiosyncratic to be even given a thought, and the fact is perhaps offer no value most major journals will not accept it due to its concepts not being attached much to something already existing. Of course, there are other papers I am referencing and citing in the paper but the concept overall is not something I think is of much interest. If it helps, the concept is developing vector spaces over the field of surreal numbers to provide a basis for studying algebro-geometric concepts in a surreal analytic framework. The journals I considered are Compositio Mathematica and AMS. 

Based on the papers I've read submitted to the aforesaid journals, they seem to prove fairly substantial results dealing with well established existing concepts. This doesn't seem to come near what I am doing since my concept is infant. 

Also, I am still in high school and am afraid of publishing. Will being a high schooler be put against me? Since I always hear if you are a high schooler your concepts be it whatever have no relevance. I am not trying to prove a major result or anything; just get my concepts out there. I've considered arXiv as an initial step to getting whatever research I have noticed but it requires sponsorship which I don't think will make its way to me.
Congratulations on developing your own research program!!

I think your observation is correct, that you may be unable to have your results published (so far). In general, papers that provide a basis for further study don't tend attract a lot of attention by themselves. 

It's not necessary need to prove an established result or problem, though. For example, if you discovered some interesting phenomena concerning algebraic varieties over the surreal numbers, for which known analogues do not exist over other fields, these could perhaps "sell" a paper.

That said, let me strongly encourage you to continue developing your theory, without worrying too much about whether other people will take interest or whether you can publish it. Intellectual inquiry, undertaken for its own sake, is valuable in and of itself; moreover, your experience will prove quite valuable if you pursue a research career in mathematics. 

Good luck!