The "early access tutorial" is just the tutorial for playing the multiplayer.
Full game is 43GB but once it's "Ready to play" at about 12GB you can go right into multiplayer on the main menu.
Ready up and start, this will put you in to the tutorial.
You'll first have to shoot some stationary objects.
Next you'll have to stand in a capture point. Then you'll have to kill actual enemies.
Then you will need to revive a fallen ally.
Then go to another capture point.
You will unlock this and Apex (below). All in all this should take you less than ten minutes and then you can go do whatever you want for the next 6+ hours you have left with the trial.
One person has reported they weren't able to start multiplayer once the "Ready to play" notification popped. So I advise starting the download as soon as possible.
This also may be one of your last chances to unlock Challenge Accepted!
in the TrueAchievement app if you still have access to snap functionality.
Thanks El Ninio GRE