I've been a tech writer at software companies for about 25 years. I got my first tech writing job with a B.A. in literature, some writing samples, and no formal tech writing education. (I'm not claiming that would be a sure bet today, just explaining my background. Unfortunately, I don't have specific courses to recommend.)
At my current company, we would be looking for the following if we were hiring an entry-level tech writer:
* Task-based writing samples. By "task-based," I mean a procedure that tells the reader how to complete a certain task.
* Familiarity with common style guides, or at least with the concept of writing to a style guide. We use Microsoft Style Guide (
https://learn.microsoft.com/en-us/style-guide/welcome/) and the Chicago Manual of Style.
* Experience with an authoring tool would be a plus. We use MadCap Flare. Adobe FrameMaker used to be a popular authoring tool.
I agree with others that writing, research, and organizational skills are more important than any specific "tech savvy" skills. HOWEVER, if you do have tech-related areas of experience or interest, those could work to your advantage. Example: My company makes software used in [industry]. If you happened to have both relevant industry experience and decent writing skills, you would be an attractive candidate to us. Industry experience is by no means a requirement, but it may be easier to break into tech writing if you look for areas of overlap between tech writing and your own personal or professional experience.
In my own work, I use HTML and CSS regularly. I don't write code, but I do incorporate code samples in my documentation. I've had to teach myself a bit about Docker and JSON files and APIs in general. I use source control (Git, Bitbucket).
I'm happy to answer any follow-up questions, if you like. Good luck!