I think you’re on a good track. I haven’t looked at the syllabus closely for our local university’s game development bachelor’s degree, but I’ve been around the net a long time. I can tell you that no matter which language you learn, once you know one well, a LOT of it can transfer to others.
The trick is learning the logic, problem solving, and math…pretty much in that order. The better you get in each, the better a programmer you will be.
Quite a lot can be done with game development on the web now. My kids, for instance, are insanely into ROBLOX and other web games. I never thought much of these types of games, but now that I’m getting into professional programming, I realized how much money is potentially there. A lot of people tend to think console is where the money is, or that you have to hit it big like Minecraft to bring in the money.
First, of course, figure out what you WANT to do, what you love. Because, whatever you choose, you’ll likely spend a lot of time doing it. If you do decide on game development, definitely keep those web skills handy. Everything is online now, even games, and I only see them merging more down the road.