There are many ways to say goodbye, each with different meanings.
Sayonara - the most common word we are given as a translation, but it more specifically means "goodbye for a while". You wouldn't say this to somebody you see every day. You usually say it when you're not going to see the person for a long time, if ever again.
Jya ne - "See you later". Very casual, usually only used in friends and family.
Mata ne - "See you again", literally "again, yeah?"
Bye bye - ...bye bye. Borrowed from English. Usually used with children, but used in the matter of Jya ne and Mata ne too.
Sarabada - Very old and very formal. It usually means "goodbye forever".