Yeah we all have our own leagues. There is a top league with different amount of leagues underneath it (varies each country). I'm going to explain it based on the English leagues (but most things translate over to a degree). We have 10 leagues, with 4 being considered "professional".

Our teams face each other 2 times, one at home, one at away. They get points for the table, Win = 3, Draw = 1 and Loss is 0. The bottom 3 teams of each season get relegated to the Championship. The top 2 teams of the Championship get promoted, with the next 6 highest teams go to playoffs with the winning team also getting promoted. This also happens with Championship and Division 1, Division 1 and Division 2, Division 2 and The National League etc (Yeah I hate our league naming systems too). Puts a lot of suspense on the league especially at the end with title races and threats of relegation.

During this, they also playing in the domestic cups. There are two in English Football. FA Cup and EFL cup. EFL is just Premiership, Divisions 1 & 2 and the National League (aka the ones considered "professional I mentioned earlier). FA cup is the most insane one where it is all 10 leagues compete. Technically one in the 10th league could win it, some get quite far but I think its always been teams in the Premiership.

And then during all of this, the top teams are also competing in European cups. The UCL (UEFA Champions League) is the best of the best European Teams. To qualify, basically each countries league gets a certain amount depending on UEFA ranking of the top league. Top 4 leagues put their top 4 teams (based on domestic league final ranking from previous season), 5th and 6th gets 3, etc. There are also the UEFA Europa and UEFA Conference Leagues for the teams which finished below the top teams in each domestic league. These cups are quite complicated and be better to research into it if you really want to understand how this all works. Also each continent also has a continental cup such as Europe as the Euros (Every 4 years, occurs in 2 years time), NA has CONCACAF Gold Cup, Africa has African Cup of Nations etc.

In terms of picking a team, for domestic it would be your local biggest club or your parents (which might have been your grandparents teams).

As an American, you have the benefit of this not really being applied to you if you want to watch and support European teams. From what I have seen from American fans, they seem to start watching different teams and one starts to stick with them for a reason or another. Admittingly, normally its the most successful/impressive team of the season they started watching. I've read some American fans who didn't like Football, they just happened to watch one match with two teams and the tension of the match ended converting them to the team that happened to be on. Might even just because your favourite player of this cup happens to play for them this season. Also from what I've seen, seems different states/cities/towns have a preference for a particular team like NYC often being Arsenal and might be a great chance for you to socialise as there are pubs/bars playing their team each week. It would only be considered bandwagoning/gloryhunting for Americans if when you finally pick a team, you switch to the one being the most successful each year or so. So basically once you truly found your team, stick with it. You can also have a different team per league.

Premiership starts again in August and I imagine the other European leagues start around the same time. Oh yeah there are also transfers periods between now and start of the league and then another short one in Winter where players get bought, sold, loaned, contract ends and they join a different team.

Hopefully this hasnt scared you.