Yeah, I guess I've never heard of freelance embedded programming :). It might be out there, but I think web is definitely the most common. Some people get into Data Science (often heavy in Python) and love it, but yeah, I think web development still has the greatest need for freelance.
When I first got serious about C# and had not done much object oriented programming, I found an older edition of what is now called "Pro C# 7: With .NET and .NET Core" to be very useful. It's massive, but large sections of the book cover large topics, and you likely won't need all of them, but you'll have a great reference that explains things. I am also a fan of Pluralsight, which is a monthly subscription to a whole host of programming/technology tutorials that all follow the same basic format. There's a lot on there, so maybe don't dive too deep at this point, but it's been great for exploring areas of programming that I maybe want to know about but am not interested in committing to learn. It's also good for those things you decide to commit to. $35 a month, I've kind of gotten my fill and may not keep it much longer, but I'd say the 3-4 months I've had it have been totally worth it.
I would advise applying some of the MMM/ERE strategizing to your goals here. What I appreciate about C#, though it applies to some other languages, too, is that it is so directly connected to so many types of programming (web, desktop, services, mobile apps, etc). It's in relatively high demand because it has the backing of Microsoft and therefore, for better or worse, more companies trust it. There have been major changes to allow it to run on other operating systems, so you can still program it on a Mac, for example, and it tends to have a better reputation than it's nearest neighbor Java (no flaming here, some people love Java and that's fine). I'm not particularly familiar with the desktop and mobile side of things, but I like knowing that with a little time I could probably build something for myself. IMO, it's a sad thing to be, say, a backend database programmer who can...well...only program databases. Which you would probably never do outside of a job situation, so it's really of no personal benefit to you, unless you simply really enjoy it. I don't know, I think you're building resilience into your system of approach if you keep these things in mind. If you had, say, 10 years of solid experience, hey study whatever you want. But your primary goal is being able to do freelance work while still being a SAHM, so it makes economic sense to appeal to the most number of employers you can to increase your chances of attaining this goal. Since your time is limited, too, it makes sense to make sure it is spent in the most versatile manner that gets you farther along on that path. Just some thoughts :)