Couldn't this also mean "I will move". Like move from Sao Paulo to Rio?
yes! (or move from Rio to Sao Paulo =) )
Yes, but since it doesn't specify, we have to suppose it means "to change".
Why? Would we need to say something like "Vou mudar de casa."?
It's usually used as a reflexive verb then:
In my experience, people always use the reflexive pronoun, or the complement "de casa/apartamento/cidade/etc".
Does this also work for changing clothes? Or behaviour?
Behaviour is the most likely meaning. Maybe appearance as well.
But for clothes, we use "eu vou mudar/trocar de roupa".
Why don't they accept shall? It's future isn't it?
That's probably because "shall" is quite an old-fashioned form for the future tense, although it's not grammatically wrong.
Vou mudar pro Brasil um dia :D :D :D
Hope you can make it =)