There are about 153,000 members of the UUA (UUA Membership Statistics, 1961-2020, uua.org). The top three states by number are: (1) California; (2) New York; (3) Texas (Demographic and Statistical Information, uua.org).
History
The UUA was formed in 1961 when the Unitarian Church and the Universalist Church merged. Let’s consider each.
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1. Unitarian Church
Unitarianism is a belief in one God which rejects the trinity. They believe that Jesus was strictly human and not deity (Frank S. Mead and Samuel S. Hill, Handbook of Denominations, p. 232). The “Holy Spirit” is understood to be used two ways in scripture. One is another name for the one God. Another refers to God’s nature which He gives man (What is the Holy Spirit? biblicalunitarianism.com). Unitarian beliefs are found in early Christianity. However, the origin of the Unitarian Church seems to be found in Protestant Reformation and later. (a) It has a European connection. “The movement spread from the independent thinkers and Anabaptist in Switzerland, Hungary, Transylvania, Holland, Poland, and Italy to England. There it found champions in such leaders as Newton, Locke, and Milton, but no attempt was made to organize the movement until the late eighteenth century” (Mead, p. 231). (b) It has an American connection. “American Unitarianism, however, developed independently, when members of the liberal wing of the Congregational Church in eastern Massachusetts, who asked that they not be required to subscribe to a creed, were branded as Unitarian.” (ibid).
The first organized church to turn Unitarian as a body was the Episcopal King’s Chapel in Boston in 1785 (ibid). A split occurred within Congregationalism in the nineteenth century. The American Unitarian Association was formed in 1825. It was a missionary society and publishing society. A national conference was established in 1865 (ibid).
2. Universalist Church
A Universalist is one who believes universal salvation. “American Universalism has its direct origin in the work of George de Benneville… John Murray… and Hosea Ballou” (Mead, p. 233). (a) George de Benneville (1703-1793) was a physician and Universalist preacher in Europe and in America, preaching in Pennsylvania and New Jersey (George de Benneville, uudb.org). He believed that the fire of hell would purify and lead to universal salvation (The Universalists: George de Benneville, reddit.com). (b) John Murray (1741-1815). He was once a Calvinist Methodist. He did some preaching in Ireland and England. He was sent to bring back a young woman who had come under the influence of James Relly, a Welsh Methodist preacher who was teaching Universalism. Murray, himself, was converted to Universalism (John Murray, uudb.org). He moved to America. His Independent Christian Church of Gloucester (Massachusetts) became organized in 1779 (Mead, p. 233). (c) Hosea Ballou (1791-1852). He was a schoolteacher and a Universalist preacher in Vermont. He too started out as a Calvinist but became convinced that Romans 5:18 taught Universalism. He published, “A Treatise on Atonement” in 1805. He also began to publish a weekly, The Universalist Magazine, in 1819. These works greatly influenced Universalists. He wrote against capital punishment and slavery (Hosea Ballou, uudb.org).
Authority
1. Sacred Texts
“While Unitarianism and Universalism both have roots in the Protestant Christian tradition, where the Bible is the sacred text, we now look to additional sources for religious and moral inspiration… we celebrate the spiritual insights of the world’s religions, recognizing wisdom in many scriptures” (Sacred Texts in Unitarian Universalism, uua.org). They do not view the Bible as inerrant (Unitarian Universalist Views of the Bible, uua.org).
2. Six Sources:
(1) Direct experience of that transcending mystery and wonder affirmed in all cultures, which moves us to a renewal of the spirit and an openness to the forces which create and uphold life;
(2) Words and deed of prophetic people which challenge us to confront powers and structures of evil with justice, compassion, and the transforming power of love;
(3) Wisdom from the world’s religions which inspire us in our ethical and spiritual life;
(4) Jewish and Christian teachings which call us to respond to God’s love by loving our neighbors as ourselves;
(5) Humanist teachings which counsel us to heed the guidance of reason and the results of science and warn us against idolatries of the mind and spirit.
(6) Spiritual teachings of Earth-centered traditions which celebrate the sacred circle of life and instruct us to live in harmony with the rhythms of nature (Sources of Our Living Tradition, uua.org).
Beliefs and Practices
1. Seven Principles
(1) The inherent worth and dignity of every person;
(2) Justice, equity and compassion in human relations;
(3) Acceptance of one another and encouragement to spiritual growth in our congregations;
(4) A free and responsible search for truth and meaning;
(5) The right of conscience and the use of the demographic process within our congregations and in
society at large;
(6) The goal of world community with peace, liberty, and justice for all;
(7) Respect for the interdependent web of all existence of which we are a part
(The Seven Principles,uua.org)
2. Jesus’ Role
They do not believe that Jesus died to save us from the wrath of God (God is viewed as too loving to be wrathful against man). Instead, Jesus is our Savior in the sense he showed us how to live (Hosea, Ballou, A Treatise on Atonement, archive.org; Tony Larsen, The Problem with Atonement, uufdc.org). Jesus lived to call us to our better selves rather than dying to save us from our fallen selves. They believe that it was Paul who changed Jesus’ role to saving man from the wrath of God (Steve Edington, Atonement and Forgiveness, fculittle.org).
3. Diverse and Inclusive
“Our beliefs are diverse and inclusive… Unitarian Universalists believe more than one thing. We think for ourselves, and reflect together about important questions” (Beliefs & Practices, uua.org).
“People with atheist and agnostic beliefs find a supportive community in our congregations… since the early 20th century, Humanism has been an influential part of our continually evolving religious traditions. Many Unitarian Universalists who are Atheist or Agnostic also identify as Humanists” (Atheist and Agnostic Unitarian Universalist, uua.org).
Organization
“Each UU congregation is autonomous” (About the Unitarian Universalist Association, uua.org). However, they do have a headquarters in Boston, Massachusetts (Headquarters of the Unitarian Universalist Association, uua.org).
Bryan Hodge 
With so many denominations and religions, how can I decide which are true and which are false?
It is certainly understandable that even a very earnest and sincere seeker after truth would be confused over the religious situation today, with hundreds of denominations, sects, and cults in Christendom alone, as well as hundreds more in other countries and cultures, and with new religious movements arising almost every day. Nevertheless, God has provided adequate instruction for us to enable us to “know the spirit of truth and the spirit of error” (I John 4:6) if we really want to do so.
There are three criteria which are especially helpful in evaluating a particular cult or movement: the teachings of its leaders concerning the Bible, concerning Christ, and concerning the way of salvation, respectively.
1. Attitude toward the Bible
The Bible claims, many hundreds of time, to be the written Word of God. The Old Testament Scriptures were accepted by Christ and the apostles as divinely inspired and completely infallible. Jesus said: “The scriptures cannot be broken” (John 10:35). With respect to the New Testament, He promised His apostles that “the Holy Ghost shall teach you all things, and bring all things to your remembrance, whatsoever I have said unto you” (John 14:26), and that “the Spirit of truth will guide you into all truth” (John 16:13).
Therefore, during the first century, the apostles who had been with Christ, had witnessed His resurrection and had received these promises, gradually wrote down the Gospels and Epistles which now comprise the New Testament. These were readily received and recognized by the early Christians as inspired Scriptures. The apostles claimed that these writings were divinely inspired and authoritative, and true Christians have always accepted them as such.
Finally, the last of the apostles, John the Beloved, near the end of the first century, was enabled to look prophetically into the future ages and to write down the last of the true Scriptures, the book of Revelation. This completed God’s written words,
“For I testify unto every man that heareth the words of the prophecy of this book, If any man shall add unto these things, God shall add unto him the plagues that are written in this book: And if any man shall take away from the words of the book of this prophecy, God shall take away his part out of the book of life” (Revelation 22:18, 19).
These last words of Christ’s apostles give us a most important rule. The Scriptures are fully inspired, even to the very words, and those who would add to them or take away from them are, to the extent they do so, false teachers.
In general, cultists have been guilty of “adding to” the Scriptures, claiming either that the writings of their own founders were divinely inspired or that the interpretations of their leaders were uniquely necessary and authoritative. Modernists and liberals, on the other hand, have been guilty of the even more serious error of “taking away from” Scripture, culling out or allegorizing those portions which they decide are unscientific or unreasonable to modern man. The true teacher, however, will accept all the Scriptures, and only the Scriptures, as the infallible Word of God.
2. Attitude toward Christ
A true Christian teacher will gladly accept and proclaim Jesus Christ as He is, true God and true man.
“Who is a liar but he that denieth that Jesus is the Christ? He is the antichrist, that denieth the Father and the Son” (I John 2:22).
“For many deceivers are entered into the world, who confess not that Jesus Christ is come in the flesh. This is a deceiver and an antichrist” (II John 7).
“There shall be false teachers among you, who privately shall bring in damnable heresies, even denying the Lord that bought [that is, ‘redeemed’] them” (II Peter 2:1).
Error concerning the person of Christ can take either the form of the ancient Gnostic heresy, which denied His true humanity, or that of the modern Agnostic heresy, which denies His true deity. The latter considers Him to be a great man and great religious teacher and leader, but rejects His virgin birth, His sinless life, His substitutionary atonement, and His bodily resurrection and ascension. Any cult or denomination or religious movement which does not clearly and forcefully proclaim the Lord Jesus Christ both as the perfect Son of man and the only begotten Son of God, “the Lord, which is, and which was, and which is to come, the Almighty” (Revelation 1:8), is false, and should be rejected.
3. Attitude toward Salvation
The gospel of Christ is “the power of God unto salvation, to everyone that believeth” (Romans 1:16). The word “gospel” means “good news,” not “good advice.” It does not tell us what we must do and not do in order to earn salvation, but rather what Christ has done to provide salvation as a free gift. “For by grace are you saved through faith; and that not of yourselves; it is the gift of God; Not of works, lest any man should boast” (Ephesians 2:8,9).
Every other religion under the sun, whether pseudo-Christian or non-Christian, panders to man’s pride by teaching him there is something he can do to earn, or to help in earning, his own salvation. Only true Biblical Christianity recognizes man as he really is, utterly lost in sin, destined for eternal separation from God. The gospel, “by which you are saved,” is the glorious news that “Christ died for our sin” (I Corinthians 15:1,3), and that we can be saved by grace, through personal faith in Christ, plus nothing else whatever! Any religion which teaches otherwise is, to that extent, false. Paul said,
“If any man preach any other gospel unto you than that you have received, let him be accursed” (Galatians 1:9).
One who is truly saved by God’s grace in Christ will, of course, then seek to follow Christ and His Word in all things, not to earn salvation, but in love and gratitude for His glorious gift of cleansing and everlasting life.
You have some good thoughts. Denominationalism is contrary to God’s plan. I am anti-denominations. Things are very confusing today. However, I do believe that it is possible to be just a Christian, and to find the church which belongs to Christ. You are correct. We cannot earn or merit salvation. It is a gift from God. When looking for the church, I would add two things to what you have said. (1) Does this church teach and practice the proper plan of salvation? One cannot be a part of Christ church without being added to it. He adds to it the saved (Acts 2:41; 2:47; 1 Corinthians 12:13). (2) Once the first point has been determined and the church is determined, I ask: Is this church conducting itself properly? e.g. Does it publicly worship correctly? How can I join in with them in worship if they do not? (John 4:24). May God be with you!