From my experience a strong portfolio is what matters most, and the piece of paper can help give you great things to put in that portfolio, but it's not mandatory for making great things. I would probably start with trying to make some things on your own first, and think about formal classes after.
I'd start by just picking up something like GMS2 or Godot and give it a try, they are free to start. Maybe see if there are any local game jams you can join too. Stuff like RPG Maker can be a good way to give it a try without getting too deep in coding. Though other 'no code' engines are not great.
In general I think it's good there are a lot of options, and there is a strong open source contender. I don't really think there is a true dominant 'de-facto' engine, every time I've been to a game jam the people I teamed up with decided on using a different engine (GMS2, Unreal, Unity) lol.
I stick to the GMS2 side of things, and that is still going strong with new devs. I think Godot's biggest drawbacks are lacking console support, and the others having more tutorials and resources from being around much longer. I think it will be a strong option, but it seems to have it's limits.
Vet art uni/classes beforehand, I took film classes at a community collage that I feel I got a lot out of. But when I transferred to a state uni for game design, they more or less lied about what they offered at my campus. It was a glorified web design class, and was a total waste of time & money.
I've had mixed experiences with formal education. Most of what I know was self-taught, but collage did expose me to things that are hard to pick up on your own, and it's good for meeting people. Ether way, making and completing personal projects on your own and building up a portfolio is important.