E.T. mimicking Elliott's behavior when they formally meet in Elliott's room. It shows that E.T. is harmless and curious.
Also Elliott's reactions throughout that scene, as he grows to care for and trust E.T. for the first time.
E.T. healing Elliott's wound.
Elliott and Michael reminiscing about their father in the garage.
The long hug E.T. gives Elliott when they're saying goodbye at the end.
Gertie's goodbye to E.T. where she gives him the flowers he revived. He tells her "Be good" and Gertie repeats what he says.
Just as E.T. boards the ship, the dog briefly runs up into the ship and then comes back down to Elliott's side.
I always thought the scene where E.T. is listening to Mary reading Gertie a bedtime story was cute.
The entirety of E.T.'s revival scene.
Elliott: Does this mean they're coming?
In the shooting script, the dialogue is different, but no less heartwarming.
Elliott: I'm taking you home, E.T. Yes. E.T. home. But first, stay. Yes? Okay? Stay. Please.
Elliott: I love you.
"I'm glad he met you first."
"I'll believe in you all my life. Every day. E.T... I love you."
Elliot reenacting The Quiet Man by planting a Big Damn Kiss on a girl in his class. At first, it's amusing and the girl is clearly taken aback, but as Eliot runs off, we get a close up of her foot popping, implying that she likes him back.
Michael is a walking moment of heartwarming after meeting E.T. When Elliot tells Michael to look for the missing E.T. he risks nearly luring the FBI, just to save him.
Prior to Gertie meeting E.T. the first words she says before opening the door are "Elliot look at what I made for you." If you look closely even when she's screaming you'll see a piece of paper in her hand. Early on Elliot was faking sick so he'd look after E.T. So Gertie, before she finds out the truth, thought he was really sick and made a get well card.