All them are common ways to say "bye" in Portuguese.
- ATÉ LOGO
It's used to express the idea of "see you soon" in all lusophone regions. Other similar expressions are:
ATÉ BREVE, ATÉ JÁ (see you soon) and ATÉ A VISTA, ATÉ MAIS (see you later).
It's used to say goodbye in a way you wish "blessings" (like the word "goodbye" in English that is a contract of "God be with you"). ADEUS sometimes is polemic because in some regions of Brazil, people see the ADEUS like a word that express the idea that you will not see the person for a long time. But in majority of lusophone regions ADEUS just mean a way to say "goodbye". Other similar expressions are:
VÁ COM DEUS (leave with God), PASSE BEM (farewell).
- TCHAU (or XAU)
It's the Portuguese version of the popular Italian word "ciao". It's an almost universal expression that you can use everytime you want to say "bye".
Another ways to express goodbye/bye/farewell in Portuguese:
ATÉ AMANHÃ (see you later tomorrow), ATÉ A PRÓXIMA (see you the next time), BOA NOITE (good night)