I know that 'who' refers to people and 'that' refers to things. But what about when the thing IS people, such as a family? Would I say, "That's the family that is moving in next door soon.", or "That's (they're?) the family who is (are?) moving in next door soon."?
You can substitute "that" for "who" in relative clauses when the clause is identifying. If the clause is non-identifying, then "that" is not permissible.