one says "on a serious note" usually when the conversation was more casual and humorous and then they suddenly want to change the topic to something more serious. See a translation.

What does it mean when someone says on that note?

Specifically when you say on that note, it means "and with that [last comment, event, etc.] we will [do something]. Let's say you are in a meeting and it is about to wind up and then somebody's chair breaks and they fall to the floor. The boss might say, "and on that note, we will adjourn the meeting."

What is meant by on a lighter note?

on a lighter note: to change the subject to a less serious topic. idiom.