The bit when Haku has just taken Chihiro/Sen to see her parents in the pig stable, and he gives her a rice ball to eat, and she starts tearing up as she takes the first bite, and then halfway through finishing, just begins bawling from all the trauma she has accumulated over the past 16 hours or so. The music at that point certainly helps.
If Chihiro crying her eyes out when it hits her that she is by herself in a strange world with only three people who actually care about her doesn't get you, the Bittersweet Ending will.
The moment where Chihiro figures out Haku's real name, and he returns to human form.
The soundtrack, man. Especially "Reprise".
She may act immature at the start, but Chihiro is ten years old, around the age when most children start to desire autonomy from their parents. She calls them 'Mom' and 'Dad' instead of 'Mommy' and 'Daddy', and she has no trouble walking off alone. However, when they turn into pigs, she runs into the street and starts desperately calling out for them because she refuses to believe that those were really her parents. It's even worse in the English dub, where she starts hyperventilating and crying, and just screams, "MOMMYYYY!" It's absolutely gut-wrenching to think of how terrified she must be in that moment.
The moment where Lin tells Chihiro not trust a thing Haku says, right after his advice lead her to sign away her name to Yubaba. Her reaction is to crouch down on the ground and start sobbing. The next morning shows that she's still miserable, and struggling not to cry since it would wake the other girls up.