Nephi's narrative technique which introduces this journey is identical to the first. In both cases, he simply states "I, Nephi, and my brethren" went on the journey (see 1 Nephi 3:9). This may be related to a grammatical construction.
Notice that the Lord provided the name of the person to whom they should make their entreaty. They were not sent on a general mission, but a very specific one - to a person no doubt prepared by the Lord to receive their message.
Why should they go to Ishmael? The following passage is from Daniel H. Ludlow's A Companion to Your Study of the Book of Mormon:
"Our present Book of Mormon does not indicate exactly why Ishmael's family was selected except that it included at least five unmarried women(!), but perhaps the following statement provides additional important reasons for this selection:
Whoever has read the Book of Mormon carefully will have learned that the remnants of the house of Joseph dwelt upon the American continent; and that Lehi learned by searching the records of his fathers that were written upon the plates of brass, that he was of the lineage of Manasseh. The Prophet Joseph informed us that the record of Lehi was contained on the 116 pages that were first translated and subsequently stolen, and of which an abridgement is given us in the first Book of Nephi, which is the record of Nephi individually, he himself being of the lineage of Manasseh; but that Ishmael was of the lineage of Ephraim, and that his sons married into Lehi's family and Lehi's sons married Ishmael's daughters, thus fulfilling the words of Jacob upon Ephraim and Manasseh in the 48th chapter of Genesis, which says: "And let my name be named upon them, and the name of my fathers Abraham and Isaac; and let them grow into a multitude in the midst of the land." Thus these descendants of Manasseh and Ephraim grew together upon this American continent.... (Erastus Snow, Journal of Discourses 23:184-85, italics added)" (Daniel H. Ludlow, A Companion to Your Study of the Book of Mormon Deseret Book Company, Salt Lake City, 1976:99-100).
Note that a forged letter from Emma Smith made another direct connection. That letter, however, is a forgery, and Elder Snow's statement is the best available information.