For 01, Yamato's establishing moment is probably in episode two, when Seadramon attacks and most of the kids are trapped on an island with no way to fight back. After rescuing Takeru, he proceeds to catch Seadramon's attention and swim away from the island.

This is the night of the first day, so apart from Takeru, all these people are just friendly acquaintances (in Taichi and Sora's case) or people he's maybe only seen in passing, and it takes him approximately one (1) day to go from 'who cares about any of you, I've got to look after my bro' to 'well, I guess I'm getting eaten by a sea serpent to save everyone. That's Just How It Has To Be Now.'

For Tri, it's probably this scene, the first half of the battle against Alphamon in episode 4.