Begone is the base level entreaty for someone to leave. To say “Fellow, begone!” is the equivalent of “Okay mate, out you go…” today. If not addressing someone who is actually a fellow, you can use any other form of address, or simply say “Begone!” with an imperative tone. A flick of the hand toward the door could add a nice dramatic touch.
Get thee gone! adds a touch of urgency. It’s more like saying “Go, quickly!” or “Get out now!” This is used forty times throughout Shakespeare’s plays, usually when there is a sense of timeliness or hurry about the leaving. It can also suggest impatience or frustration with the person to whom the command is addressed.
Get thee hence! is equivalent to “Get out of here!” or “Get away from here!” It often seems stronger and more urgent than begone! or get thee gone!
It is certainly expressive and delivers a satisfying sense of Shakespearean drama to your demand to be left alone. Of course, if they don’t go when you tell them to, you can always try mixing it up a little just as Imogene did in Cymbeline: “O, get thee from my sight… Dangerous fellow, hence!”
Aroint thee! Is stronger again because of its implied disrespect for the recipient of the command.It really just means ‘go away’ or ‘begone’, but at the same time indicates that the speaker holds higher status or demands more respect than those to whom they are speaking. It also has supernatural or spiritual connotations, as it was commonly used to eject witches from one’s presence.
Avaunt takes telling someone to get lost to another level, as it carries even greater spiritual or superstitious weight. This is the word one would use for commanding demons or any other evil presence to leave. When directed at people rather than the supernatural, it carries connotations of derision, hatred, or fear; that the speaker seeks to protect themselves from those to whom they speak is understood from this choice of word.
It is important to note that a great deal of suspicion toward witches existed at the time at which both plays were written and performed, and at both points in time in which each play was set. Therefore, Shakespeare’s use of injunctions such as aroint thee and avaunt is a clear indication that the characters on the receiving end are either held in contempt and/or malevolent.

A longer, more detailed exploration of each of these terms is available at Shakespeare Nerd.
Shakespearean Ways To Tell Someone To Go Away.
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Okay. Now I need to find a way to insert “Get thee gone!” into a conversation. Hmm. Perhaps I’ll try it on the next person who interrupts me when I’m sleep into a book.😁
It’s remarkably common in my life!