Blizzard say that "most" FTUE restrictions will be lifted when playing in a group. So the quickest way to get your hands on as much of Overwatch 2 as you can will be to team with friends.
However, this won't be the case for Competitive mode, because players will have to complete a specific challenge before they're allowed into this game mode. Instead players will have to win 50 Quick Matches before they're allowed into Competitive.
Anyone who has played the original Overwatch prior to the October 4 date won't be bound by these restrictions, nor will those who purchase the Watchpoint Pack.