If you use "get out" you are unhappy with the situation: "I need to get out" --you are bored or frustrated or in danger where you are and you want/need to leave to make things better "I want you to get out" --someone is bothering you and you want them to leave, or this person is in danger and you are quickly telling them to leave where they are. "I want to go out" --you are going to do something because you feel like it, you are in a good mood, and you would still be happy even if all of a sudden you couldn't leave. "I need you to go out" --you are politely asking someone to leave, either to do you a favor or to lend you privacy, but either way you are being polite to them and this is not a bad situation. The difference is subtle, and tone of voice matters a lot here. Typically, you calmly say phrases with "go out" and you shout or have an angry tone when you say "get out"

Ex. We are going out of town. let's go out (lets go do something). Are you going to go out tonight? Get Out!!!! ( Say that when you are angry and you don't wan't somebody in your presence). Lets get out of here. (lets leave here). Get out your money. (telling you to take your money out.)

"go out doing something" is like "go out while doing something", "go out in a certain manner" described by the specific 'something' in the sentence. "go out to do something" is like "go out in order to do something", "go out for the purpose of doing something". Note that in the first construction "go out" can be a literal going out or a figurative (idiomatic) one. Literally, "The detective was so angry that he went out muttering to himself": He left the premisses; he was muttering to himself at the time. "to go out" = "to leave", "to exit" Figuratively, "The team won the final game of the season. It was quite a thrill (for them) to go out winning": They finished/ended the season as winners. "to go out" = "to finish (a season, a career, one's life)". Also, again figuratively, "The poor guy finally died of cancer, but at least he didn't go out whining about the injustice of life": He didn't die (bitter and) whining. "to go out" = "to die". The second construction is normally literal: "Let's go out to get some fresh air". "They've often gone out to get ice cream at midnight".

Hello Tatyana: We can go out to a ballett (where you are going to) We can get out of our apartment. (where you are coming from) Also, in anger, if you want some one to leave (Get Out!)

get out = change of state/position from being in to being out go out = many meanings. Try a dictionary. But you don't use it when you specifically indicate the change of position as described above.