Do most major math journals not care about a piece of research unless it proves an established result or problem?
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!