They travelled for many years, and had various experiences.
In chapter 18 of 1 Nephi in the Book of Mormon, they finally reached the land that was promised to them by the Lord.
I love how Nephi credits the Lord in every step of the journey. In verse 3, he tells us that he did go oft unto the Lord; wherefore the Lord showed unto me great things.
Lehi and His People Arrive in the Promised Land
I, Nephi, did guide the ship, that we sailed … towards the promised land.
And … after we had sailed for the space of many days we did arrive at the promised land; and we went forth upon the land, and did pitch our tents; and we did call it the promised land. (1 Nephi 18:22–23)
“Yes, this shows the ship that Nephi built. Nobody knows what his ship looked like. All we are told is that it was not built after the manner of men. . . . I don’t think God would instruct Nephi to build some very weird thing never seen in heaven or earth just to prove that it had divine help. It would be some perfectly sensible principle of shipbuilding that was perhaps in advance of what was known to shipbuilders at that time.
“This moment is when, with great relief, they finally sighted land, so for the moment the fighting between them is forgotten in the excitement of seeing land.
“The birds are not seagulls, but rather swallow- tailed roseate terns, which are found in the tropical waters around Central America. Such details helped define the geographic location for this painting. Lehi is looking heavenward in thanks, while the other guys are pretty much like in a pirate picture, shouting ‘Land Ho!’ The huge ropes were from the movie The Ten Commandments, and they were brought from Egypt. The Bedouins there weave these immense ropes by hand.” 
I love reading the Book of Mormon.
And if ye shall believe in Christ ye will believe in these words, for they are the words of Christ, ... and they teach all men that they should do good. (2 Nephi 33:10)
Seriously, just go here.
If you wish to order a free copy, let me know or click here.
Feel free to read or listen to it here.
 Arnold Friberg, as quoted in Vern Swanson, ‘The Book of Mormon Art of Arnold Friberg, “Painter of Scripture”, Journal of Book of Mormon Studies: Vol 10, 1, 26-35, Provo, Utah: Maxwell Institute, 2011
[Image found at http://store.lds.org/webapp/wcs/stores/servlet/Category3_715839595_10557_21096_-1___0]