This question is vague and has some loaded terminology ("not yet a famous person"?), but the question overall is good, so I will answer a closely related question, which is:
Where is a good place to present new mathematical results if I am unknown to the broader mathematical community?
I'll write about the US context, since that is what I know about.
In that context, I'm surprised that no one has suggested simply registering for the next AMS Sectional Meeting near you and submitting an abstract for the general contributed paper session. While it is not super likely that an "expert" in the result will attend your talk, it is a good chance to give a (short!) talk about your result in a low-stress environment. And the Sectionals are pretty cheap. My view is that simply gaining experience giving a concise talk is more valuable than trying to get big press for something. The publication, should things work out, would be how to get cognoscenti to pay attention. But writing a good paper and giving a good talk are quite different skills.
If you are fortunate and your area is one of several that often have people outside of "professional mathematics" contribute, you could also submit an abstract to one of their sessions at a Sectional (or the Joint Meetings), though here there is not always room. But there are certainly some subfields, especially some interdisciplinary ones, that encourage these submissions. Be aware that here the issues raised in the other answers will definitely come into play if it is clear you aren't fairly immersed in the literature.
Finally, in some areas (especially of "recreational math") the Math Association of America meetings will have sessions that are more research-focused and open to newcomers.
If you are not based in North America, I'm less sure whether there are similar events. Perhaps there might be joint meetings between 'advanced-high-school' and 'beginning postsecondary' studies folks? In any event, I would recommend the local equivalent of a regional conference that has at least one 'no-rules' generalist session and costs less than the PPP equivalent of $100 to attend.
Naturally, it's always caveat lector when it comes to "solving famous problems", so one has to be careful in the situation posed by the poster. Don't present a solution to the Goldbach Conjecture publicly without having someone who knows what they are doing vet it! Instead, give a talk about the GC and its history, and then in the second five minutes (yes, it could be that short) say, "here is a new approach that shows promise". Maybe someone in the talk will point out an obvious error. Or maybe it will turn out it really is a promising new approach. Or maybe you already solved it. But you won't have claimed to prove the GC and then have to retreat.