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.

I have a few suggestions which may be field specific.

First, check out arXiv. It is sometimes used as a place to "park" research prior to peer-reviewed processes such as journal or conference submission. Some authors (who are usually already "big names") publish there and accrue citations, too. Although, as pointed out in comments, you may require some academic input to pass through arXiv's requirements to post. There’s also technically nothing to stop you "publishing" on your own website (or github!), but your work may receive little attention there.

Second, you must be sure that your contribution is novel. Absence of evidence is not necessarily evidence of absence and all. The reason you might get a desk rejection is because it’s too easy to dismiss an author with no affiliation and therefore unlikely to have access to a sizeable body of literature. Journal subscriptions can be fairly expensive. If you’re affiliated to an institution, you often get access to many journals that you wouldn’t pay for as an individual. There’s still a reasonable number of publications that aren’t open access. You might get some access via library membership (worth checking). An (appropriately) extensive literature review with citations should overcome this barrier.

Third, journals may publish your work with no costs to you, but you will probably have to pay for open access. That means you’ll sign over copyright of your work and, if you don’t have a subscription to that journal, potentially be unable to see it (unless you paid for open access - fees I’ve seen are 3-4 figures).

Fourth, you should be aware that some journals want you to recommend reviewers. There are some questions on here regarding that already, so I won’t go into the politics of that.

It’s worth getting an academic on board simply to navigate the world of academia and increase the chances of getting published. Alternatively, can you patent it? That would protect your idea even if you choose not to profit from it.