Matthew talking Marilla into keeping Anne, and Marilla buckling down. But Matthew better not stick his oar in!
Oops, he does, by giving Anne her first pretty dress for Christmas. With puffed sleeves, no less!
Gilbert leaving a candy heart on Anne's desk to apologize for the carrots incident.
In Anne of Ingleside, she has a necklace he gave her that looks like a candy heart. This was given to her back in Anne of the Island by Gilbert. She wore it to a party instead of the pearls her boyfriend Roy had given her. Or at least she would have if a misunderstanding hadn't ensued.
Gilbert finding and keeping one of Anne's flowers that she lost at the concert.
Anne and Matthew's last conversation together.
"I'd rather have you than a dozen boys, Anne. Just you remember that."
Anne deciding to stay at Green Gables and teach, because of Marilla going blind.
Gilbert then switching teaching jobs with Anne, so she can be closer to Green Gables to help Marilla.
Not entirely a moment, but Marilla coming to love Anne. If there is a moment here, then it is Marilla finally admitting her love to Anne.
The scene when Anne speaks with Diana's rich Aunt after they accidentally disturbed her in bed. By the time Anne is through with her, that old woman is completely charmed by the girl and requests that Anne visit her because she is amusing, "And at her age, amusing people are a rarity."
The book ends with a scene between Anne and Gilbert, in which they resolve their difference (or rather, Anne does) and they become friends.
In "Anne's House of Dreams" they discuss this scene on the eve of their wedding. Gilbert confesses that "all heaven opened up" before him and that he was the happiest boy in the world as he left her at the gate of Green Gables.
Anne of Avonlea
Anne trying and then succeeding in getting through to Anthony Pye.
Marilla is discouraged because she can't get Davy Keith to behave, but Anne thinks there's still hope: "Remember how bad I was when I first came here." Even though Marilla spent much of the first novel reprimanding Anne for various things, she says now that Anne was never bad; granted, Anne got into plenty of scrapes, but her intentions were always good.
Anne of the Island
Anne's Heroic BSOD after hearing that Gilbert was dying. Yes, it's cliche. Yes, the monologue where she realizes she was in love with him the whole time is trite. But darnit, seeing the usually-poised Miss Shirley, pride of Avonlea, brought down to a shambles by one slip of the tongue of a child just shows how masterful L.M. is at making us feel what the characters feel.
How about Gil finally getting the girl he patiently waited ten years for?
The poem at the opening of the book that foreshadows the above events from one of Anne's favourite poets, Tennyson.
All precious things, discover'd late,
To those that seek them issue forth,
For love in sequel works with fate,
And draws the veil from hidden worth.
It's one of the most heartwarming excepts ever (It's quoted from Tennyson's The Daydream) and so perfectly exemplifies Anne and Gilbert's relationship in this book.
Anne's House of Dreams
Anne and Gilbert's wedding.
When Anne and Gilbert arrive in their new home, Gilbert introduces Anne to Captain Jim. It's the first time he's introduced her as his wife to anyone, and he "narrowly escaped bursting with the pride of it."
Chapter dedicated to James Matthew's birth. Especially when Gilbert announces to Marilla (who has, understandably, been very nervous about this event) that Anne is doing fine after the delivery and that the baby is healthy: "He weighs ten pounds and — why, listen to him. Nothing wrong with his lungs, is there?"
That whole chapter is just full of hilariously adorable moments, mostly provided by the supporting characters. There is Anne's anger after being told Jem's hair will be red, Susan telling Anne and Gilbert that they needn't ever be ashamed of the baby's ears, outrageous stories about what/how families in the Glen named their children, and Miss Cornelia's story about her brother—she had always wanted a brother and eventually got one, but was horribly disappointed to find it was a younger brother; she'd wanted a brother two years older than her.
Anne of Ingleside
7 years old Walter walking several miles alone in night to get to home, after he heard that his mother was dying. It turns that Anne was giving birth to Rilla