Learn Portuguese (Portugal): Separo tudo para a mudança.

Tudo means "everything" and is an invariable pronoun, meaning it never changes to match masculine, feminine, or plural. You use it when talking about things in general. You only use todo, toda, todos, or todas when referring to a specific noun, like todas as coisas (all the things).

Do we ever need to put an article with tudo, like o tudo?

No, tudo stands completely on its own. However, when you use the variable forms to mean "all of the" or "the whole", you usually do need the article right after them, like in toda a mudança (the whole move).

Doesn't mudança usually mean "change"? How does it mean "the move"?

It means both! The literal translation is indeed "change" (from the verb mudar, to change), but in everyday Portuguese, a mudança or the plural as mudanças is the standard way to talk about moving from one house to another.

Why use the preposition para here instead of por?

We use para here because it expresses a goal, purpose, or destination. You are separating everything with the specific goal of the move. You would use por if you were talking about the cause or reason for an action.