July 24, 1847 was when the Mormons arrived to Salt Lake City. Prior to that they migrated from New York, to Kirtland Ohio, to Nauvoo, Illinois, and Missouri. Each time they moved was due to the people not wanting them there, and thus they were hunted down and persecuted everywhere they went within the United States. They decided to leave to another nation. Mexico, at the time, owned Utah. Unfortunately for the Mormons, the US took Utah from Mexico very soon after they arrived. Fortunately, they were unharmed for the most part by Americans. The fact the US did eventually take Utah, though, does mean that they were subject to American Law again, so when the US passed the Morril Anti-bigamy law in 1862, it applied to them since Utah was part of the US. Thus, a couple of decades later the US started to imprison Mormons for practicing bigamy/polygamy.
Anyways, my last two sentences up there aren't really relevant to Mormon Migration, per se.
So, once in Utah, the church started to do a lot of Missionary work. Well, actually they were before that, but it picked up a lot more. They converted people primarily from Northern Europe at this time. Those newly-converted Mormons migrated to the US and to Utah. They came from ships, then took the railroad as far west as it went, but it didn't go all the way to Utah. From there, they may have taken caravan, or, in many cases, they had to do hand-cart. The Willie and Martin handcart companies are the most well-known hand cart companies. This was mostly because those immigrants came into Missouri(that was the last state that railroads entered, they went no further than western Illinois), and they left late into the season. I think it was August that they started to cross the Great Plains. Since it takes a few months to cross the plains into Utah by hand-cart, they ended up going into wintertime. The terrible snow slowed them down(which was a problem since they didn't bring food to count for being slowed, so they had to give very low rations), and it would cause death by hypothermia in many cases. So, in short, the Willie and Martin handcart companies had the highest death rates of the any other crossing of the plains Mormon immigrants did.
Some key terms to know:
"Mormon war": during their time in Missouri, it was misery for the Mormons. Many disputes took place and much fighting between Mormon militia and Missouri's militia took place.
Haun's Mill massacre 1838: Colonel William Jennings was a leader of a militia which attacked a group of Mormons after the fact that they called for a truce at Haun's Mill.
Missouri Executive order 44: Governor Boggs of Missouri believed the Mormons were causing too much trouble for the state of Missouri and ordered them to be extradited from the state or else be killed. Thus, he made it legal to kill Mormons in Missouri and this order wasn't removed until the middle of the 20th century.
Joseph Smith Jr: Leader of the Mormon church, author, proprietor, and translator of the Book of Mormon, considered the Mormon prophet, and became mayor of Nauvoo, Illinois. He ran for the office of president of the US in 1844 since when he went to the president before, the president basically ignored his complaints that the Mormons were being persecuted and not protected by any of the states they resided in. So, to remedy that, he attempted running for president to be able to fix this issue himself. He never got to really campaign much since he was put in jail well before election day and was killed.
The Nauvoo Expositor: This was a newspaper company which was in Nauvoo, as one would expect. They printed material critical of Joseph Smith's doctrines of Plural marriage and exaltation. Former-mormons and non-mormons made up the employees and owners of the Nauvoo Expositor. Joseph Smith, as mayor of Nauvoo, ordered their printing press to be destroyed. Due to this, he was put in Jail in Carthage on June 25, 1844, killed on June 27, 1844.
Carthage Jail: this is where Joseph Smith, his brother Hyrum, John Taylor, and Willard Richards were sent to Jail after the destruction of the Nauvoo Expositor's printing press. Joseph and Hyrum were both killed on June 27, 1844 when an anti-Mormon mob marched into the jail and began firing at them. John Taylor and Willard Richards survived, however. John Taylor later became the Mormon Church's third prophet and leader, by the way.
Brigham Young: during Joseph Smith's leadership over the church, he was the head of the Quorum of the 12 apostles. The quorum of the 12 is comparable to the importance of Cardinals in the Catholic church, but not exactly the same. Anyways, after Joseph Smith's murder, Brigham Young became the leader and new prophet of the church. There was some controversy on who should become the head of the church. Many, including Emma Smith(Joseph's first wife) thought Joseph Smith Jr's son, Joseph Smith III should have been it, and they broke off from the original church and became the Reorganized Church of Jesus Christ of Latter-day Saints. Majority stayed with the main church, since I think they saw the logic of the head of the quorum of the 12 succeeding since Jesus supposedly made the head of the quorum of 12 he had(Peter) as the head of his church after himself. Jesus didn't make his church succeed from father to son, as the Reorganized Church(it's known as the Community of Christ today) did. Anyways, so Brigham Young became Prophet, and when he did in 1847, he lead the Mormon exodus to Utah to seek solitude and safety for the Mormons, since their previous prophet was murdered and they had to deal with years of persecution to begin with.
Succession problems: As I mentioned before, there was trouble with who should become prophet after Joseph Smith. A total of 5 people, besides Brigham Young, claimed succession. Joseph Smith III, had the second largest following, Brigham Young having the largest. Sidney Rigdon had the third, and then the others had such a small amount, they're not worth mentioning. It took a total of almost 3 years for this to finally be settled(it started in 1844, but Brigham Young didn't become the prophet until 1847), and even then, each of those other people who claimed succession decided to split off and form their new church.
Willie and Martin handcart companies: in 1856, James G. Willie and Edward Martin formed handcart companies. They started in England, and got Mormons who were currently living in England, Denmark, and Norway, across the Atlantic to the USA. They took a train up until around Nauvoo Illinois where the railroad ends, and they pushed handcarts the rest of the way to Salt Lake. They left late in the season, and you can see the rest of what I said above.