What level of math does one usually require before being able to publish a paper of their own
This is something that varies with every individual. The best advice is to first get a broad mathematical education and then focus on some area of interest. But along the way to that broad education, seek insight into why things work as they do, not just the computational skill that you normally get in something like Calc 2.
Some people are able to publish as undergraduates, but it is fairly rare. However, an undergraduate math degree should get you to the point that you have actual insight into one or two areas (mine were analysis and topology) sufficient to dig deeply.
Research is normally very narrow, focusing on some tiny aspect of some small subfield if mathematics (or physics, or ...). But before you go narrow, first go broad, so that you have some sense of the possibilities. And also, some sense of the interconnections between things.
Working with a professor on some project of mutual interest will give you both a sense of what it is like to explore the unknown and also lets you take advantage of their insights. And letting them know now of your goals can help you attain them.
Solve a lot of problems. Get a lot of feedback. Look for the why, not just the how. Notice things. Keep notebooks. Ask questions.

This is, I know, more than you asked for, but you seem eager to learn and I appreciate that. Also note that whatever &quot;research&quot; I did as an undergraduate was, though attempted vigorously, laughably naive. Oh well. It came, eventually. And it was &quot;hella fun&quot;.