Building up the kingdom and establishing Zion

 Kingdom:  

1. A contiguous territory or country under the dominion of a king.  

2. The inhabitants or population who are subject to a certain king or queen. “The whole kingdom rejoiced.”

Zion: 

1. A specific geographical location variousely transliterated Sion, Tzion, Tzion, Tsion, Tsyyon, is a placename in the Hebrew Bible used as a synonym for Jerusalem, as well as for the Land of Israel as a whole. You can find it in 2nd Samuel

2.A group of people who are pure in heart.  "Therefore, verily, thus saith the Lord, let Zion rejoice; for this is Zion--the pure in heart; therefore, let Zion rejoice, while all the wicked shall mourn. “  

Spencer Kimball’s thoughts on Zion and how we create it:

"Zion is 'every man seeking the interest of his neighbor, and doing all things with an eye single to the glory of God.' (D&C 82:19.)

"First, we must eliminate the individual tendency to selfishness that snares the soul, shrinks the heart, and darkens the mind.

"Second, we must cooperate completely and work in harmony one with the other.

"Third, we must lay on the altar and sacrifice whatever is required by the Lord. We begin by offering a “broken heart and a contrite spirit.” We follow this by giving our best effort in our assigned fields of labor and callings. We learn our duty and execute it fully. Finally we consecrate our time, talents, and means as called upon by our file leaders and as prompted by the whisperings of the Spirit. 

"My brothers and sisters, if we can do this, then we will find ourselves clothed in the mantle of charity “which is the greatest of all, for all things must fail—

“But charity is the pure love of Christ, and it endureth forever; and whoso is found possessed of it at the last day, it shall be well with him.” (Moro. 7:46–47.)

"Let us unite and pray with all the energy of heart, that we may be sealed by this bond of charity; that we may build up this latter-day Zion, that the kingdom of God may go forth, so that the kingdom of heaven may come. 

From "Becoming the Pure in Heart" , Spencer W. Kimball, April 1978

It seems that “building up the kingdom of God and establishing Zion” is simply the work of building up (strengthening) the people who recognize God as their king and are pure in heart and, with them, establishing groups of people who seek to love and respond the way Jesus does.