The game to get better at is the game you enjoy most, and what sites depends on the game.

This is spot on, though keep in mind that, depending on the game, you might play it so much you’ll grow to hate it. You’ll also find yourself looking at videos and guides on how to get better, effectively upping you from a casual

Small time or local competitions/tournaments are likely to only have one mr. motherfucker that will utterly destroy you. They tend to show up a lot more in fighting games, though some strategy games, like Starcraft 1/2, also have a significant number of them. If you want to avoid feeling humiliated, racing games (Gran Turismo, Forza, Project CARS) are the ones you’ll want to go for. Sports games (football, 'murican football, basketball) also have really damn good players, though it’s unlikely you’ll get humiliating losses in them after a bit of training.

As someone else mentioned, speedrunning is also an option, though it’s a lot less about direct confrontation and more about grinding the same game over and over. It’s also the easiest one to “get into”, especially if you aim at less known games. Events like AGDQ can hold head-to-head competition of 2 or more runners in the spot.

Keep in mind that some games need teamwork as much as individual skill - Marvel Rivals, Overwatch 2, League of Legends, etc.