My verdict: The movie kind of dragged. The main conflict, which is supposed to tie the film together, was actually what I found to be least appealing. They have Brigham Young lie about being the next prophet. Despite his personal doubts about his ability (or right) to lead, he repeatedly deceives the people into believing he was a prophet. But this concept of Brigham Young has no historical basis. When he finally decides to "come clean" he prays to the Lord. Unfortunately he is standing in front of a portrait of Joseph Smith when this happens, so it looks like he is praying to the picture, not to God.
I also wasn't fond of the subplot involving the young LDS man, Jonathan, and his romantic interest in the non-LDS girl, Zina. It didn't really contribute to the plot. They also invented a character (Angus Duncan, based, in part, on the real-life character of Sidney Rigdon) to be the personification of all of the opposition Brigham Young experienced. But he prevaricated so much as a character that I couldn't take him seriously.
Their treatment of LDS theology was a bit spotty. They allude to polygamy  a few times, but never really address it. They focus a lot on the socialist aspects of the United Order  without addressing the essential theological components of it. The miracle of the gulls was kind of rushed.
Other observations I had: The actress who played Emma looked like she was in her fifties, and the soundtrack, by Alfred Newman , appropriately incorporates some LDS hymns ("The Spirit of God" and "Come, Come Ye Saints") but also inexplicably includes some odd choices ("Oh! Susanna").
 Joseph Smith, Jr. was assassinated by a Missourian mob on June 27, 1844. The film was released on September 27, 1940.
 The portrayal of the assassination of Joseph Smith was a little inaccurate, but the alterations were forgivable. The young Vincent Price, who plays Joseph Smith, looks surprisingly normal and gives a reasonable performance as Joseph Smith. In fact, in a letter written to James d'Arc (curator of the motion picture archives in the Harold B. Lee Library Special Collections at Brigham Young University) he mentioned studying for the part with President Heber J. Grant and his admiration for Joseph Smith and the Latter-day Saints.
 To learn more about the Church, I suggest you visit here and here.
 I think Dean Jagger resembled this portrait of Brigham Young a lot. Thirty-two years later Dean Jagger was baptized into the LDS Church.
 Supposedly Zina was introduced to act as a foil for the audience, helping them to sympathize with the Mormons. I'm unaware whether this ploy was effective, though I suspect not since this film wasn't successful anywhere—not even in Utah.
 Polygamy, also called Plural Marriage, was practiced by certain members of the Church at this time, including Brigham Young. The practice was discontinued on October 6, 1890. It is now only practiced by splinter groups that are unaffiliated and unsanctioned by the Church. Polygamy is currently grounds for excommunication from the Church.
 The United Order was a brief experimentation with communalism. The Wikipedia article on the United Order (here) gives a passable description and highlights differences between it and communism.
 See http://en.wikipedia.org/wiki/Miracle of the gulls.
 Alfred Newman has the second highest number of wins at the Oscars (behind Walt Disney) and is tied with John Williams for the second highest number of Oscar nominations (again, behind Walt Disney).
Covered Wagon at Scotts Bluff is by J. Stephen Conn, available at http://www.flickr.com/photos/83372564@N00/4151180201.