What steps would an amateur scientist have to take in order to get a scientific breakthrough published?
The title question is answered by "write a paper and submit it to a journal". The amateur scientist doesn't need anything special to do that.

The questions in the text deal with a separate question, which is whether or not the breakthrough actually is a breakthrough. In this case getting a professional opinion is certainly going to be helpful (see Kaveh's answer to a related question for the process). It's easy to deal with the "threat" of the professor stealing the idea: just establish precedence by, e.g, attaching the manuscript to an email with a timestamp. If the professor tries to steal the idea anyway, he would be breaking some deep-rooted academic norms. If it's proven that he's plagiarizing, he can get into serious trouble.