To go away is to completely leave for a long period of time. To go out is to get out of the place where you were out. To leave is to completely get out of the place you were at and to go somewhere else.

You could use either in this instance, but "leave" is the more accurate term of the two: "Go" refers to the whole journey whereas "leave" is the act of moving away from the current location - the beginning of the journey. leave is used in reference to the place I am moving from, and go to the place I am moving to.

In the example, it describes regular or frequent activity... in that sense we also use "go away" as "travel". Examples of this: To go away on vacation. We are going away on vacation. He always goes away on vacation in the summer. To go away for the weekend: We are going away for the weekend. He often goes away for the weekend.

go out means: go to walk go away means: getting somewhere else leave means: It's just leaving somewhere.