I wrote something similar about Destiny back in the day, but here is how I've always imagined the different levels of PvP structure to work:
Beginner Level
This starts with character familiarity. Mystery Heroes is a great place to get your feet wet. It's not about know specfics, it's more about picking stuff up like which characters are faster, and which are slower. Which ones have fast shooting guns, which ones have slow. Which ones have offense abilities, healing, defensive, mobility abilities. Because character interaction is the root of this game, and there are so many characters, this is the baseline of all future interactions and advanced skills.
Knowing every characters kit is a good sign you've climbed beyond a beginner.
Novice Level
This is where map familiarity starts to come into play. Every map has a unique geometry to it, on top of different game types and of course health pack locations. The way the characters interact with that geometry will start to become important. You'll begin to understand where widows and hanzos hang out. Where Rein's like to setup. Where Roadhogs like to stalk. Where Reapers like to perch.
You'll begin to develop a bit of muscle memory regarding map geometry once you've reached a certain point. Like, Gibralter's first checkpoint is coming up, and the attackers have a Reaper on their team. So you start reflexively looking on top of that corner structure to see if he's going to drop an Ult soon.
Amatuer Level
Counter plays are going to start coming into play. Now that you've become familiar with the characters and the maps, you've started getting a sense for which characters seem to be able to win against specific enemy characters. Now you might find yourself switching character to better counter the enemy team, or to create better synergy.
You've started to learn that Tracer can charge her Ult crazy fast against an enemy Roadhog. You've started to learn that Winston can chew up an enemy Hanzo that's too far away from his teammates. You've started using Ults to counter other Ults, or Abilities to stop Ults dead. You've decided to play a Zarya to combo with the Hanzo on your team. Or an Ana to work with the Rein and DVa combo. You've even found yourself switching to a different character even though you've had your Ult ready to go for 2 full minutes but couldn't find the right time to use it.
You may also find yourself trying to peak at the enemy team before the game officially starts to you can better pick your character to go out the door with.
Expert Level
Muscle Memory is taking over fully at this point. You see a Roadhog hook and you know you've got a window to move in on him before he can 1HKO you or a teammate. You see a Lucio use his Ult, so you count to 5 and then activate your Tactical Visor. You see the enemy Zen go Transcendance, so you wait a sec and then charge him as Rein so that by the time his Ult is over he'll hit the wall shortly.
Map geometry is second nature at this point, as is damage intake. You step out of a door and get hit from 10 o'clock. Not only do you know almost exactly where the enemy is shooting you, but you can tell from the pacing of the bullets exactly who it is. And knowing that information you have already made the fight of flight call.
You're running a mental clock on how the enemy team Ults are looking. Putting pressure on the more lethal characters in hopes they're use a Ult in a poor situation. You're trying to force the enemy team to split up so that you can attack them without a healer. You can glance at the clock and know that you've got 3 more solid attempts at caping the objective before the round is over. You know that even though you've killed 4 of the enemy team members, you're hesitating to kill the 5th simply because it would force more time out of them to regroup later on.
Obviously there are more elements at play here, but I feel like this is a decent summary of the different levels of skill.