Brigham Young, The Teachings of President Brigham Young, Vol. 3

I can say that it is good and wise and judicious in parents to instruct their children in the way. If they wish the word of the Lord upon the subject, I will give it to you and you may, any of you, write it down if you please. It is the will of the Lord our God that we teach our children the way of righteousness from the Holy Scriptures and there is no better method than for mothers to teach them at home, and in the Sunday Schools.

Joseph F. Smith, Masterpieces of Latter-day Saint Leaders, comp. N. B. Lundwall, 1953, p. 78

I would like my children, and all the children of Zion, to know that there is nothing in this world that is of so much value to them as the knowledge of the gospel as it has been restored to the earth in these latter days through the Prophet Joseph Smith. There is nothing that can compensate for its loss. There is nothing on earth that can compare with the excellency of the knowledge of Jesus Christ. Let, therefore, all the parents in Zion look after their children, and teach them the principles of the gospel, and strive as far as possible to get them to do their duty—not mechanically, because they are urged to do it, but try to instill into the hearts of the children the spirit of truth and an abiding love for the gospel, that they may not only do their duty because it is pleasing to their parents, but because it is pleasing also to themselves.

I found this quote while reading an Ensign Article from Elder L. Tom Perry called "Therefore I Was Taught," May 1994

Spencer W. Kimball, President Kimball Speaks Out On Planning Your Life, pg 91

This mortal life is the time to prepare to meet God, which is our first responsibility. Having already obtained our bodies, which become the permanent tabernacles for our spirits through the eternities, now we are to train our bodies, our minds, and our spirits….We have this life of limited years in which to learn of God, to become the masters of our own destiny and secondly, we have this life plus eternities to learn of the earth and the things thereon, and to accumulate secular knowledge which will help make us gods, which is our destiny.

This quote can be found at this link

Gordon B. Hinckley

Behold your little ones. Pray with them. Pray for them and bless them. The world into which they are moving is a complex and difficult world. They will run into heavy seas of adversity. They will need all the strength and all the faith you can give them while they are yet near you. And they also will need a greater strength which comes of a higher power. They must do more than go along with what they find. They must lift the world, and the only levers they will have are the example of their own lives and the powers of persuasion that will come of their testimonies and their knowledge of the things of God. They will need the help of the Lord. While they are young, pray with them that they may come to know that source of strength which shall then always be available in every hour of need.

The full article for this quote is found here. It is definitely worth the read!!

Joseph Fielding Smith, Doctrines of Salvation, Vol. 3, pp. 203-4

Children should be encouraged in the home to read and be instructed in the value of good books and how to discriminate between the good and the bad in literature.

Doctrine and Covenants 88:77-80

77 - And I give unto you a commandment that you shall teach one another the doctrine of the kingdom.

78 - Teach ye diligently and my grace shall attend you, that you may be instructed more perfectly in theory, in principle, in doctrine, in the law of the gospel, in all things that pertain unto the kingdom of God, that are expedient for you to understand;

79 - Of things both in heaven and in the earth, and under the earth; things which have been, things which are, things which must shortly come to pass; things which are at home, things which are abroad; the wars and the perplexities of the nations, and the judgments which are on the land; and a knowledge also of countries and of kingdoms-

80 - That ye may be prepared in all things when I shall send you again to magnify the calling wherunto I have called you, and the mission with which I have commissioned you.

Alvin R. Dyer, "Education: Moving Toward and Under the Law of Consecration," 1969

I think that by the end of the millennium, for those who will occupy the celestial kingdom, the home will be the only media of teaching children. Teaching will be through the family.

David O. McKay, April 1968

Education to be complete must include spiritual growth. In this sense, youth need religion. I shall offer only three reasons this evening for giving proper religious training to youth.

• First: Youth should have religion in order to stabilize society. Goethe has rightly said that "the destiny of any nation at any given time depends on the opinions of its young men under five and twenty."
• Second: Youth need religion to satisfy the innate longing of the soul. Man is a spiritual being, and sometime or another every man is possessed with a longing, an irresistible desire, to know his relationship to the Infinite. He realizes that he is not just a physical object to be tossed for just a short time from bank to bank, only to be submerged finally in the ever-flowing stream of life. There is something within him that urges him to rise above himself, to control his environment, to master the body and all things physical, and to live in a higher and more beautiful world.
• Third: Youth need religion to comply properly with the purposes of creation. There is a purposeful design permeating all nature, the crowning event of which is man. Here, on this thought, science again leads the student up to a certain point, and sometimes leaves him with his soul unanchored. For example, evolution's theory of the creation of the world offers many perplexing problems to the inquiring mind. Inevitably, a teacher who denies divine agency in creation, who insists that there is no intelligent purpose in it, undoubtedly impresses the student with the thought that all may be chance.

There is much more to this quote and the entire talk can be found by clicking here.

James E. Faust, Ensign, Nov. 1990

As societies as a whole have decayed and lost their moral identity and so many homes are broken, the best hope is to turn greater attention and effort to the teaching of the next generation—our children. In order to do this, we must first reinforce the primary teachers of children. Chief among these are the parents and other family members, and the best environment should be in the home. Somehow, some way, we must try harder to make our homes stronger so that they will stand as sanctuaries against the unwholesome, pervasive moral dry rot around us. Harmony, happiness, peace, and love in the home can help give children the required inner strength to cope with life’s challenges.

Boyd K. Packer, Charge to the David O. McKay School of Education, October 1996

In many places it is literally not safe physically for youngsters to go to school. And in many schools - and it's becoming almost generally true - it is spiritually unsafe to attend public schools. Look back over the history of education to the turn of the century and the beginning of the educational philosophies, pragmatism and humanism were the early ones, and they branched out into a number of other philosophies which have led us now into a circumstance where our schools are producing the problems that we face.

John W. Taylor, Collected Discourses

There is a spirit working among the Saints to educate their own offspring. If our children will be all we will have for a foundation of glory in eternity, how needful that they be properly trained... There are wolves among us in sheep's clothing ready to lead astray our little ones... Wolves do not devour old sheep when there are any young ones. I have herded sheep long enough to know that. Look after your children.