"Get in" is like a command. It has a sense of urgency to it. It's raining outside, you're walking on the street. I drive up in my car, roll down the window and tell you, " get in!" (Hurry! Get inside!) "Go in" is more of an invitation. You are waiting to see the doctor in the waiting room. The nurse calls your name and says, " you may go in". Like, "please step inside". Enter is more of a written form of "go in". You will see it written on signs, doors, etc. It can also be said, it's just more instructional.

In addition to what Жанна had explained, "enter" is also a form of a command from a superior. You knock on the door of your boss' office. He can say either "come in" which is a friendlier and warmer while "enter" is more formal command.