However, from my experience it does not always lead to a fulfilling career. I know too many people that follow this path thinking it is smooth sailing once you land a job, only to later be bored out of their minds (they refuse to even look at a line of code outside of work). Ask yourself this, are you doing it to be a 'software engineer/ game designer', or are you doing it because you have a passion for coding and all the things that come with it?
In my opinion, the best way to get into these fields is to find a problem you'd like to solve in your life and code up a solution for it. It doesn't have to be big. My first one was adding a loading bar to a website for $50 (enough to buy a newly released video game). It was arguably my worst paying job since it took me over 100 hours to figure out how to add 20ish lines of code to a static site. It was also one of the most euphoric moments in my life. I refreshed that website with my newly added loading bar at least a couple hundred times haha.
Anyways the TLDR is:
School will give you a solid foundation of knowledge that can get you a job. Being curious will give you a life long tool to solve any problem that comes your way.
To quote Steve Jobs:
“You’ve got to find what you love. And that is as true for your work as it is for your lovers. Your work is going to fill a large part of your life, and the only way to be truly satisfied is to do what you believe is great work. And the only way to do great work is to love what you do."
My short career summary:
Degree in Biology -> Educational researcher in neuroscience -> CTO and Co-Founder at a multi million $ startup -> Software engineer at IBM -> Software engineer at Google
In this time I've done dozens of projects outside of work for fun (and profit!).
I am listing a few communities where you can create your profile and get tons of information while searching by specific topics: https://github.com/, https://www.gamedeveloper.com/, https://www.gamedev.net/start-game-development/, https://www.linkedin.com, https://developer.microsoft.com/en-us/games/community/, https://medium.com/, https://www.pluralsight.com/.
I hope this helps. Good luck!