If I think I have a publishable idea on a certain problem, how do I publish and proceed?
You are on the cusp between math and CS, actually. The better way to "publish" your work is to write it up formally, with references/citations and everything and submit it to either a journal or a conference. Choose one that matches the content of your paper. 

But also make sure it is a reputable journal and not one that just publishes anything for the money you pay them.

However, you also have to be prepared, with journals, for publication fees, even with reputable ones. Sometimes they may waive the fees, but you may need to find a way to pay them. And, for conferences, make sure you have some way of attending if your paper is accepted for presentation. In CS, conferences are usually preferred. In math, usually journals.