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 garbage.
The long hug E.T. gives Elliott when they're saying goodbye at the end.
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?
"I'm glad he met you first."
"I'll believe in you all my life. Every day. E.T... I love you."
While Elliot reenacting The Quiet Man is a moment of funny. The girl reacts to his kiss like she is taken aback by it makes it heartwarming because it implies she likes Elliot.
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.