What is the most common phrase for wishing good things for for people about the weekend? I thought about something like "have a nice weekend" or "have a sweet weekend" etc.
Have a good weekend or Have a nice weekend are by far the most common idiomatic ways to say this. Any other option is so much less common that it's notable just for being different.
You can technically use any adjective that means "good" or "enjoyable" - have a great weekend, an awesome weekend, a fantastic weekend - but sweet is not an idiomatic replacement for good; if someone said "Have a sweet weekend!" to this native US English speaker, it would strike me as distinctly unusual.