I've never seen a Mormon establish doctrine on the Book of Mormon, only arguing that the Book of Mormon is "another testament of Jesus Christ." So, since Mormons believe that is the case, surely they base some doctrine on the book of Mormon. So what important doctrines do Mormons believe are taught in the book of Mormon that we couldn't find in the Bible already?
First, I am slightly surprised that you've never seen Mormons use the Book of Mormon to teach/establish doctrine. (Talk to the missionaries; I bet they'll use the Book of Mormon quite extensively.)
This was a bit tricky to think of at first, since the Bible does indeed mention the main, important doctrines. Because of that, the Latter-day Saints use the Bible often in their studies and in church. However, there are many doctrines/principles are explained in the Book of Mormon that the LDS believe are not clear from the Bible alone:
The Apostasy and Restoration. The Bible does refer to an apostasy and eventual restoration, however, it is vague at best. The Book of Mormon clarifies that the Priesthood keys and fulness of true scripture will be taken from the Earth for generations, then restored through heavenly messengers in the last days to prepare for the second coming of the Savior. It even describes that a continent (America) will be set apart and preserved for the Restoration. (1 Nephi 13)
The Fall. While the Bible rehearses to us primary events of the placement of Adam and Eve in the Garden of Eden and their subsequent choices, it isn't entirely clear why their choice to eat the fruit was important, or whether it was good or bad. The Book of Mormon drills the point home that the Fall of mankind was an essential part of God's plan: fully anticipated, with a redemption through the Atonement of Christ prepared for the salvation of mankind. (2 Nephi 2).
Life After Death. The Book of Mormon explains that death is not simply the transition from mortality to the eternal heaven/hell. We lose our physical bodies at death and enter the spirit world, where we remain until the resurrection and judgment, at which point, we eventually enter the kingdom prepared for us. (Alma 40; 2 Nephi 9).
The Purpose of Christ's Suffering in Gethesemane. The word "Atonement" appears dozens of times in the Book of Mormon, but at least it does show up once in the Bible. It is the central event of all the ages, and without it, we would be lost forever. The Book of Mormon clarifies this over and over, and helps us understand what really took place in the Garden of Gethsemane, and the far-reaching effects of His having completed the Atonement perfectly. (Mosiah 3; Alma 7)
The innocence of children. The Book of Mormon teaches that little children need no repentance, nor baptism, and that baptizing children is an evil abomination. Children are alive through the Atonement of Christ, and children who die before an age of accountability will be saved. (Moroni 8)
The Nature of God. The Bible documents the doings of God quite extensively, and the Book of Mormon more clearly explains His character and disposition. For example, in Ether 12, Moroni is comforted by the Lord who teaches him that weakness is a boon for coming closer to Him and remaining humble: that the Lord gave us weaknesses so He could draw us to Him and make us strong.
We are saved by grace after all we can do. Different Christian religions believe different things about the requirements for salvation. The Book of Mormon helps us understand that it is grace that saves us, after all we can do, and the Book of Mormon emphasizes that good works are necessary, but not sufficient, to gain salvation. We must truly become a new being through the Atonement of Christ. (See these passages; Mosiah 27; Alma 12)
The meaning of baptism. The Book of Mormon explains why the sinless Christ was baptized beyond simply "to be born again" -- and why each of us needs to follow that example. (2 Nephi 31)
The importance of religious freedom. The Book of Mormon teaches about good leadership and government, especially as it pertains to the free practice of religion. Latter-day Saints and the Church are thus strong advocates for religious freedom. (Mosiah 29)
Again, many of these are touched on in the Bible, but not wholly or completely enough to be unambiguous or not leave significant room for interpretation (hence the many different Christian denominations). But off the top of my head, these are the primary doctrines that are the least clear or mentioned in the Bible and the most clear and discussed in the Book of Mormon.
As you have said, we believe the Book of Mormon to be another Testament of Jesus Christ. It is a second witness to what the Lord has established. In 2 Corinthians 13:1 (KJV) we read:
This is the third time I am coming to you. In the mouth of two or three witnesses shall every word be established.
The Book of Mormon establishes and offers that second witness, and often clarification to what is taught in the Bible. There are no doctrinal differences.
- It offers the second witness of Christ's baptism, and explains what it means to "fulfil all righteousness".
- We are taught more concerning the Fall of Adam and Eve
- It expounds on baptism, and especially infant baptism
There are plenty more examples, but I digress from the question. We do base our doctrine off of revelation found in the Bible, the Book of Mormon, other scriptural canon, and current revelation from a living prophet. All of this, however, does not deviate from what Christ has taught.
The fundamental doctrine that was not included in Matt's excellent post is that there is a "Plan of Salvation." While many of the elements exist in the Bible, the acuality of a plan by that name (or its variants) cannot be found in the Bible.
Quoting from the Book of Mormon, the three following verses discuss the plan of salvation:
…For what could I write more than my fathers have written? For have not they revealed the plan of salvation? I say unto you, Yea; and this sufficeth me.
And the great God has had mercy on us, and made these things known unto us that we might not perish; yea, and he has made these things known unto us beforehand, because he loveth our souls as well as he loveth our children; therefore, in his mercy he doth visit us by his angels, that the plan of salvation might be made known unto us as well as unto future generations.
For behold, if Adam had put forth his hand immediately, and partaken of the tree of life, he would have lived forever, according to the word of God, having no space for repentance; yea, and also the word of God would have been void, and the great plan of salvation would have been frustrated.
Another important doctrine is the Restoration of the Gospel and that God calls prophets today. Neither one of these doctrines are written in the pages, per se, but they are a result of the book's existance. The logic is:
Therefore, the only problem the objector has to resolve for himself is whether the Book of Mormon is true. For if the Book of Mormon is true, then Jesus is the Christ, Joseph Smith was his prophet, The Church of Jesus Christ of Latter-day Saints is true, and it is being led today by a prophet receiving revelation.
—Ezra Taft Benson, President of the Quorum of the Tweleve Apostles, April 1975 General Conference, "The Book of Mormon Is the Word of God"
Another part of doctrine which is not found in the Bible but is discussed at length in the Book of Mormon is the concept of agency, that is that we were intelligences before the world was created, and we are "agents unto ourselves" and more specifically, we are able to "act for ourselves" This freedom to act is also fundamental to our answer to "the Problem of Evil."