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).
- ADEUS
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".
P.S.:
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)