These are all different ways of walking, and they are difficult to explain with words.
But perhaps I can clarify a few of the words.
Stagger = this is how a man walks when he is drunk on alcohol, or perhaps he is seriously injured.
"Swagger" is more often used as an adjective instead of a verb these days.
It means something like "masculine confidence."
"To prance" = this was used in the past to describe how a horse walked in a parade or a performance. When this word is used to describe a human, it means that they are walking in a way that is boastful. Most of the time, we use this word as a joke.
"To sashay" = this is to describe the way a fashion model walks in a fashion show. Usually the shoulders and hips move in an exaggerated way.
Good luck!