We’re going to leave the very formal goodbyes here and dive into the familiar – or overfamiliar – ones.

Bye… would be “Salut !”
Ciao… would be “Ciao”, “Tchao” (pronounced the same way as Ciao… all’italiana 😉
See ya… would be “À toute” or “À plus” (short for À toute à l’heure and À plus tard)
See ya later… would be “À plus” or “À plus tard !”
So long… would be “À la prochaine” (meaning see you next time)

I will not explain the situations when you should or should not use them… it’s about the same in French and English 😉