Below is a timeline of the evolution of Visiting Teaching. Dates and facts have been taken from the Encyclopedia of Mormonism, Vol. 4, Visiting Teaching, unless otherwise noted. It is quite short, and elaboration that anyone can give will be added.
Relief Society founded.
1842, March 24
Second meeting of the Relief Society. Emma Smith suggested appointing persons to wait upon the poor.
1843, July 28
Necessity Committee of sixteen was named “to search out the poor and suffering, to call upon the rich for aid.” The two original functions were: “to ascertain the condition of the families visited, and to accept contributions for charitable purposes”
In the early years of the Church in Nauvoo, Illinois, visiting teachers reported their visits at the regular Relief Society meeting before all members present, citing specific instances of need.
Following the martyrdom of the Prophet (Joseph Smith), the Relief Society was discontinued for several years (Teachings of Presidents of the Church: Brigham Young, p. 131).
Relief Society established in some Utah wards. (Teachings of Presidents of the Church: Brigham Young, p. 131)
With the arrival of Johnston’s army from the United States, Relief Society, along with other ward organisations was discontinued. (Teachings of Presidents of the Church: Brigham Young, p. 131)
President Young called on the bishops to reestablish the Relief Societies. (Teachings of Presidents of the Church: Brigham Young, p. 131)
Visiting Teachers were relieved of the personal responsibility of both ascertaining and meeting the material needs of families, but since then they have continued to report confidentially (to the ward Relief Society president) any instances of illness or need requiring attention.