When you say farewell or goodbye in a formal way, you can also shake hands, just as you might when you greet someone.
Depending on the context, if you want to be very polite you can also say Vielen Dank (thank you). For example:
Vielen Dank für Ihre Hilfe – when someone has helped you
Vielen Dank für den netten Abend – when you've had a nice evening together
This is the most common way of saying goodbye. These days it is used throughout Germany.
Most people say goodbye informally in various ways:
Ciao! – especially popular among young people
Bis bald! – when you will see someone again soon
Wir sehen uns! – what friends say to each other when they see each other regularly
Mach's gut! – a relaxed way of saying goodbye when you want to wish the other person luck
Einen schönen Tag noch! – a polite goodbye, especially used in shops
Auf Wiederhören! – a formal goodbye on the telephone
Servus! – In southern Germany and Austria, it's not only used to say "hello", but also to say "goodbye"
Tschö! – in the Rhineland in western Germany