Does HN have any advice on how to get started with synthesizers, with an eye towards creating a game soundtrack?
https://learningsynths.ableton.com/
For a more advanced set of lessons there is Syntorial, which has a demo including the first 20 or so lessons and the full set of 200 lessons for $120. Syntorial requires a download, but it's a full synthesizer with lessons built into it, so you can use it in your own music.
Aside from that I'd say youtube is the best source. While you're still a beginner I'd recommend avoiding any videos where they don't show the finished result because it's much harder to tell if the video is BS or not.
For free synthesizers, I'd recommend "TAL U-NO-60" as a good beginner one, it's modeled after the Roland Juno 60. And "Vital" for a more complex / less beginner friendly one. Both of those have paid versions with additional features but those additional features are not necessary at all. You'll also need a DAW software capable of loading VST plugins to use them.