According to Mormonism, how does a person get saved? My question is pretty straightforward; I'm just wondering how a person gets "saved" according to Mormons.

By "saved" I'm referring ultimately to entrance into the "third heaven" of Mormon doctrine, which as I understand it, would correspond to the place that Chalcedonian Christians call "Heaven" (i.e. the "New Earth" of Revelation 21).

I know Mormonism teaches that all non-Mormons will go to "a heaven", but as they are considered apostate, they don't make the cut for getting into "the third heaven" (i.e. what the rest of us mean when we say "heaven".)

So what does a person have to do to make the cut? If the answer varies based on gender, race, etc., please indicate this in your answer.
The simplest answer to this is given in the Articles of Faith:


  3 We believe that through the Atonement of Christ, all mankind may be
  saved, by obedience to the laws and ordinances of the Gospel.
  
  4 We believe that the first principles and ordinances of the Gospel
  are: first, Faith in the Lord Jesus Christ; second, Repentance; third,
  Baptism by immersion for the remission of sins; fourth, Laying on of
  hands for the gift of the Holy Ghost."


An additional principle made clear in scripture but not cited in this summary is that it is necessary to "endure to the end" in exercising faith, repenting,  and obedience to the commandments of God. The LDS consider salvation to be a process more than an event, a process which is usually not complete until one has died. (This is why they typically have difficulty giving a simple "yes" or "no" answer to the question "Have you been saved?")

Race and gender do not matter. The degree of accountability or responsibility  a person has does. In addition,

Doctrine and Covenants  137:7-10


  "All who have died without a knowledge of this gospel who would have
  received it if they had been permitted to tarry, shall be heirs of the
  celestial kingdom of God. Also, all those who shall die henceforth,
  who would have received it with all their hearts,  shall be heirs of
  that kingdom. For I, the Lord, will judge all man according to their
  works, according to the desire of their hearts. And I also beheld that all 
  children who die before they arrive at the years of accountability are saved
  in the celestial kingdom of heaven".


Most references in the scriptures to salvation refer to salvation in the celestial kingdom.  Two other degrees, and the requirements for each of these degrees, are described in Doctrine and Covenants section 76. There is also a state of no salvation described there.

Faith in Jesus Christ is understood to include acceptance of his latter-day prophets and apostles (Joseph Smith and successors) and the additional Testament he has given (the Book of Mormon) as well as the Old Testament and the New Testament. Anyone may repent and all are commanded to, but repentance must be genuine and involve a permanent change of conduct, not lip service only. The ordinances of baptism and the laying on of hands for the gift of the Holy Ghost must be performed by those who have been ordained to the same priesthood given to Joseph Smith and according to the order of the Church.