I'm wondering where I should flick to in my LDS quad in order to learn about what Jesus supposedly got up to when he visited America. Could someone provide an outline of his interactions on the American continent(s) along with scriptural citations? I'll be happy to hear from anything in the LDS corpus of scripture.
If you have a quad, [a four-in-one book containing all four Standard Works that Mormons recognize as canonical scripture, for those unfamiliar with the term,] this account can be found in the Book of Mormon, in 3 Nephi, beginning at chapter 11. It contains an account of Jesus's ministry among the Nephites, (the inhabitants of the land who the Book of Mormon is a history of,) how he preached the Gospel to them, established/reorganized the church under the Law of the Gospel instead of the Law of Moses, and performed miracles among the people.
The people in the Book of Mormon had a prophecy among them that when Jesus is crucified there will be a great destruction in the land followed by three days of darkness. When it happened and during the time of darkness the people heard Jesus' voice which spoke to them. This is recorded in 3 Nephi 9 and 10 (https://www.lds.org/scriptures/bofm/3-ne/9?lang=eng).
One thing that most don't realize the importance of is that after calling The Twelve among the Nephites (and they were NEVER called apostles but just referred to as "the twelve" though they filled an equivalent role) and setting in order some important points of doctrine, Jesus gave a discourse that is very similar to the Sermon On the Mount. The significance of these discourses is that they contain the new commandments that go along with the New Covenant. It is in Matthew 5:3-16 that the commandments are given and then they are identified as being commandments in verse 19. In the discourse given to the Nephites Jesus was more pointed in identifying that those equivalent verses (3 Nephi 12:3-16) are, indeed, the commandments for the New Covenant:
"19 And behold, I have given you the law and the commandments of my Father, that ye shall believe in me, and that ye shall repent of your sins, and come unto me with a broken heart and a contrite spirit. Behold, ye have the commandments before you, and the law is fulfilled. 20 Therefore come unto me and be ye saved; for verily I say unto you, that except ye shall keep my commandments, which I have commanded you at this time, ye shall in no case enter into the kingdom of heaven." (3 Nephi 12:19-20)
The fulfilling of the Old Covenant and instituting of the New Covenant is very significant in religious history and that is one reason why Jesus personally delivered it to the Nephites. That is also why both sermons (the one given to the Jews and the one given to the Nephites) have the same message with only minor variances.