If by "more serious" you mean more mechanically complex than "point and click", I recommend trying arena shooters if you're willing to get your booty cheeks slapped into the dirt for a bit while learning to play. Quake Champions is free and has the best GFX, but it has hero shooter aspects that a lot of people don't like. Quake Live is a paid (but cheap) alternative with no bullshit. (And I'd recommend the original Quake if it weren't so dead...still might be fun with friends, though.)

Half-Life 1's multiplayer is my favorite and it recently got a 25th anniversary update, but tbh it's still a chaotic and broken mess lol. But if you're into that, I highly recommend it.

I would recommend Counter-Strike as a more straightforward yet still mechanically complex FPS, but they recently removed a LOT of content. Lots of game modes and maps are missing, but if you're in it for the competitive experience you might still like it.