“Hasta pronto” means literally “see you soon”. You say it when you expect to see the other person in a short period of time, usually defined (for example, if you’ll meet again later in the day).
“Hasta la vista” means “so long”, and it’s used when you don’t know when you will meet the other person again.
The difference between “hasta luego” and “adiós” is less clear. Both refer to a short lack of time until the next meeting, but some people think that “hasta luego” is more polite. However, in Spain is very common to alternate both when leaving: if you say “hasta luego” the other will say “adiós” and vice-versa. It’s just a habit.
As you’ll see in future lessons, being polite in Spanish it’s many times not a matter of the words you say, but HOW you say them. A kind a sweet “adiós” will always sound more polite than a rude “hasta luego”.
¡Hasta pronto!
