Introduction
Competitive TF2, like any sport has two main parts. Formal play, competing in a league against teams with your very own and informal play which include PUGs (Pick Up Games). You can think of informal play as sparring or playing ball at a community centre. It's meant to be lower stress, with lower stakes and a different configuration of rules. Many people participate in more informal play because there is a lower commitment.
The two main spheres of interaction are league play and PUGs. For North America, the main league that people play is RGL (Recharge Gaming League). I recommend you play in RGL if you want to actually play on a team. PUGs nowadays are relegated to discord servers, some more public than others. I will provide some recommend places for you.
There are many different competitive formats but by far the most popular in order are: 6's (6v6) and Highlander (9v9). For 6's, you may have seen that the primary roster of classes is 2 scouts, 2 soldiers, 1 demo and 1 medic. 5CP and KOTH are the primary game modes. Specialists or "off-classes" are used in more specific scenarios that are more befitting. I.e, engineer for defending last point, pyro for stuffing ubers or breaking certain holds and sniper for getting a pick. Highlander forces representation for all classes and use KOTH, Payload and Attack / Defend as its game modes. My recommendation is that you play 6's, in my opinion it's much more fun, has less moving parts on the surface, there are more resources to help you and will give you a more calm room to learn fundamentals. Judging by your gravitation towards Demoman and Soldier this shouldn't be too much of an issue. Highlander can be fun but I think 6's is better for beginners.
Practical Advice
First, play Newbie Mixes on Fridays 9 PM EST. They run games that have coaches sit in to teach you the game. The pace is slower, more strict and they pause after every round to go over what went right or wrong. Since they sit in, they can also provide you feedback on your communication.
Next, since Newbie Mixes is only on Fridays you may want to play more PUGs or just games at different times. You should start with TF2 Coaching Central; then branch out from there.
If you enjoy playing 6's and want to get on a team, you should sign up for the Newbie Mixes Team Drive. They manually matchmake other players into a 6's team for RGL so that you can participate in league play alongside a coach to guide you. The current RGL season is half-way in so the Team Drive will probably start around late spring. This is an 8 (and potentially more if you make playoffs) week long commitment where you will have one official match-game on Wednesdays and scrims throughout the week. If you want to be part of a team with less commitment you can signup as a substitute player who fills in when the main starter is unable to.
Outside of the program, finding a team is a process of talking to other people or responding to LFT / LFP posts in the RGL discord.
TF2CC also hosts the Newbie Cup 2-3 times a year. This is an internal event where they in a similar vein, manually match make you into a team with a coach. You train for 3-4 weeks and then play official matches over the following weekend. There are other events that you can engage in but they don't realistically come by that often.
I hope this is useful and you become a familiar face around the scene. Cheers.