You have asked some good and far-reaching questions.
In brief, the United Church of God observes the festivals outlined in Leviticus 23 because they were commanded by God in the Old Testament and that command was never abrogated in the New Testament. In fact, Jesus Christ observed them, as did His apostles and the New Testament Church long after His death (Luke 22:15; 1 Corinthians 5:6-8; Acts 2:1; 18:21; 20:16). We should follow their example (1 John 2:6; 1 Peter 2:21; 1 Corinthians 11:1).
In introducing His command to keep these festivals, God makes an important point: "These are my appointed feasts, the appointed feasts of the Lord, which you are to proclaim as sacred assemblies" (Leviticus 23:2, New International Version, emphasis added throughout).
Most people assume these feasts are only for the Jews. Yet God clearly tells us they are His feasts, not the feasts of any particular ethnic or religious group. Further, He says they are His "sacred assemblies"—appointed times or divine appointments when He commands His people to gather before Him.
One of the major flaws in mainstream Christianity today is how the Old Testament or Hebrew Bible is generally viewed. Over the centuries there has been an enormous effort to cut Christianity off from its roots by claiming that much of the Old Testament is no longer applicable or necessary to the Christian life. Yet the New Testament itself emphasizes the importance of the Old Testament.
Consider what the apostle Paul wrote to Timothy: ". . . From childhood you have known the Holy Scriptures, which are able to make you wise for salvation through faith which is in Christ Jesus. All Scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness, that the man of God may be complete, thoroughly equipped for every good work" (2 Timothy 3:15-17).
There was no New Testament canon when Timothy was a child. The "Holy Scriptures" to which Paul referred were what is today commonly called the Old Testament. Far from being obsolete or unnecessary, Paul says these words are "given by inspiration of God" and "profitable for doctrine, for reproof, for correction, for instruction in righteousness, that the man [or woman] of God may be complete, thoroughly equipped for every good work."
Marvin Wilson, professor of biblical and theological studies at Gordon College, Wenham, Massachusetts, lamented the general state of Christian knowledge of the Old Testament in our modern age:
". . . Many Christians seem to have little knowledge about their biblical roots [in the Hebrew Bible]. They have never really penetrated the inner world of biblical thought. Christians can converse intelligently about the latest automobiles, fashions, music and sports, but too few give evidence of a deep understanding of their spiritual heritage. At best, their grounding in biblical soil is both shallow and shaky . . . Churches have conveyed the attitude that a thorough understanding of this [Old] Testament . . . is, more or less, optional for today's Christian" (Our Father Abraham, 1989, pp. 4-5, 108).
The New Testament mentions Jesus, the apostles or members of the early Church observing six of the seven annual feast days. Only the Feast of Trumpets is not mentioned. But the event that this particular annual Holy Day heralds—Jesus Christ's return—is mentioned prominently in the New Testament.
What few people realize is that these biblical festivals teach us the major steps in God's plan for the salvation of mankind. Passover, for example, foreshadowed the sacrificial death of Jesus Christ as the Redeemer of mankind (1 Corinthians 5:7)—which took place on the very day of the Passover. The Feast of Unleavened Bread depicts the necessity of Christians to remove sin from their lives and live a new, sinless life in Christ (verses 7-8).
The Feast of Pentecost, also called the Feast of Firstfruits, foreshadowed the founding of Jesus Christ's Church, made up of those who are the firstfruits in God's plan of salvation (James 1:18; Revelation 14:4). And, not coincidentally, God founded that Church on the Feast of Pentecost when He sent His Spirit to the 120 disciples gathered in Jerusalem (Acts 2). In every sense these are truly Christian Holy Days, as they revolve around Christ's role in God's plan.
If you would like to read more about all of the annual biblical festivals and their meaning, we suggest you read our booklet God's Holy Day Plan: The Promise of Hope for All Mankind.
Observance of the Holy Days does not obligate us to keep the law of sacrifices today. Indeed, since the destruction of the Jerusalem temple in A.D. 70, there is no Levitical priesthood to offer them or a place designated by God at which they can be offered.
Even if we did have a temple and a priesthood, sacrifices would not be required for Christians today because the one great sacrifice of Christ made the need for animal sacrifices redundant during this Church age (Hebrews 9:11-14; 10:1-14). This is one of the major points expressed in the book of Hebrews.
Animal sacrifices were part of the administration of the affairs of the physical nation of Israel, whereas today the Church is a spiritual nation called "the Israel of God" (Galatians 6:16). The apostle Peter made it clear that the Church is "a spiritual house, a holy priesthood, [called] to offer up spiritual sacrifices acceptable to God through Jesus Christ" (1 Peter 2:5).
Those sacrifices include our prayers to God as well as the presentation of our lives as a living sacrifice (Romans 12:1). This latter expression refers to a life that is dedicated to living God's way and keeping His laws.
This is a rather broad subject, and this answer has hit only some of the high spots. Please be sure to request the free booklet God's Holy Day Plan: The Promise of Hope for All Mankind. Also ask for Jesus Christ: The Real Story, as it will make plain Christ's teachings on the necessity of observing God's law.