So let’s start with…
Yes, leagues (plural).
Overall there is Major League Baseball (MLB)
MLB is broken into the American League and the National League.
The biggest difference between the two is the previously discussed DH rule. The American League has it 
, the National League does not 
So there is your first “weirdness”–a major sport with two leagues that play under slightly different rules.
Every season, the two leagues face off twice.
About the middle of the season there is the All Star game, where select players from both leagues (selection is partially based on fan voting) play National League vs American League. Unlike the NFL’s All Star game, this game is fairly serious. It only results in “Bragging Rights”, but it is a big deal.
At the end of the season, each league has a play off to determine the league champion. That team wins “the pennant”; literally a pennant (i.e. saying “2019 National League Champions”) that is flown over their home field starting the next season. Then these two teams play in the World Series, a best of seven games contest. The winner is MLB’s Champion for the year.
I’ll look into web broadcasts. The good news is that (I’m pretty sure) most of the games are played in the day.
A word of caution. Baseball is referred to as the American “Pastime”, not “Sport”. The action is not nearly as intense as American foot ball. It is a game of statistics and often subtle tactics based on those statistics.
For example: I have seen a right handed pitcher replaced to put in a left handed pitcher (known as a “South Paw”) because, statistically the player at bat has a poorer “batting average” against left handed pitchers. (Batting Average is the ratio of base hits to official times at bat…more on that later)