All are correct and used in casual situations like among friends. Their nuances in English are じゃあ; See you. また(ね); See you soon. じゃね; See you. (じゃあね is more natural) You can also say それじゃ(あ); See you then.
When you say "See you" to elder people in Japanese, "それでは(失礼します)" is the best. It also means "See you".