Do Mormons believe that there is no salvation outside of their church?
Yes, they do. Mormons believe that the Church of Jesus Christ of Latter-Day Saints is the church that restores Christianity to the form it originally had in the times of the apostles.
Mormons believe that soon after Christ’s death, the apostles themselves were killed, and the direct earthly line of authority from God ended. Christianity ceased to exist with the death of the twelve apostles.
Because divine authority had been lost, Christianity began to drift away from the original church set up by Christ. Men who had good intentions but without the proper authority from Jesus Christ began to lead the church away from the truth. The Catholic Church is entirely corrupt. The Reformation, spearheaded by Luther, Calvin, and others, tried to reform the church to the truth, but they too failed.
Then in 1820, God the Father and Jesus Christ came to Joseph Smith in a vision and instructed him to restore the true Church of Jesus Christ.
Do Mormons believe that a person has to believe in Joseph Smith?
Yes, they believe that Joseph Smith is the greatest prophet. Anyone who has any concern for the eternal well-being of their soul must pay heed to Joseph Smith. Everyone who has lived since the days of Joseph Smith MUST accept him as God’s true prophet in order to enter the eternal presence of our heavenly Father.
Where does Brigham Young fit into all of this?
Brigham Young concurred with this teaching. In 1859, in his Journal of Discourses, Young wrote:
From the day that the Priesthood was taken from the earth to the winding-up scene of all things, every man and woman must have the certificate of Joseph Smith, junior, as a passport to their entrance into the mansion where God and Christ are—I with you and you with me. I cannot go there without his consent.
Mormonism differs sharply from biblical Christianity. Some of their teachings are novel—such as the belief that the Garden of Eden is located in the State of Missouri—but other teachings can only be considered heretical and dangerous. Mormonism doesn’t differ from the biblical revelation in just a few minor points. It is a different religion.