To say goodbye formally in German, say "Auf wiedersehen." This literally means "until we see each other again." You can use this when you end a meeting with somebody. How do you say it? It sounds like "owf veeder zehn" in English.
At the end of a telephone call, say "Auf Wiederhören," which sounds in English like "owf veeder huhren."
This literally translates into "Until we hear from one another again." This is contextually correct since you cannot really see someone again on the phone.
At the end of a day either in person or on the telephone, say "Gute Nacht." This sounds like "goo teh nahkt" in English.
To say goodbye casually, try "tschuess," which sounds sort of like "juice" but closer to "chooce." It is a derivative from other language alternatives such as "adieu" in French, and its literal translation is "God be with you."
Say "Bis Spaeter" (which sounds like "bis spay tah") to say, "See you later" in German. There are several ways to say goodbye in a more casual context. You can also say "bis dann" or "bis bald" (until soon). Try "bis Morgen" when you will see each other the next day.