Why should God require us to worship him?
Is worship degrading?
Some people object that worship is degrading and therefore God shouldn't require it. Yet what is worship? Worship is not groveling before God, nor is it performing meaningless rituals. God hates hypocrisy and insincere worship (Hos 6:6, Is 1:11-17). What God does tell us to do is to love him (Mt 22:35-40). Loving God includes praising him, thanking him and obeying him (1 Jn 5:3). If God is perfect, then he deserves praise, and his commands are good commands that should be obeyed. Worshiping God is not degrading to us, but actually benefits us:
Worshiping God fulfills us and makes us happy. We like to see that a person's goodness is acknowledged and rewarded; praising God gives his followers the same satisfaction.
Worshiping God puts us into the proper relationship with him, in the sense that we are acknowledging that he is God and deserves worship, and doing what he wants us to do. (Being completely right with God requires more than worship - see The Four Spiritual Laws.) This allows us to have a closer and more fulfilling relationship with God (see Do people need God? for more on the benefits of knowing God).
By worshiping God, we acknowledge his perfection, which is the first step in learning from him: we realize he's a perfect teacher, and then we are willing for him to teach us and increase our wisdom and morality.
Publicly expressed worship is one way that people find out who God is. When they come to know God, they too will find fulfillment and grow into better people, which benefits not only them but society at large.
Is God vain?
Some people object that God doesn't merely accept worship, he demands it. They picture God as an egotistical or even insecure tyrant who insists that everyone tell him how great he is. This is not an accurate portrayal, for God's command to us is, "Love the Lord your God with all your heart and with all your soul and with all your strength" (Dt 6:5), not "Tell me five times a day how wonderful I am." Worship is included as part of God's command to love him, for it's a proper expression of our love for someone who is perfect and so much above us in every way. If it's fitting for us to praise our friends and family when they do well, how much more appropriate it is for us to praise a perfect God! When we love God and realize how awesome he is, worship and praise are natural results.
God's instruction to worship him is only a demand in the sense that God's other moral laws are demands. God doesn't command us not to murder because he's a dictator, but because it's morally right (and therefore ultimately in our best interests). Similarly, God tells us to worship him because it's the proper way for us to relate to him and because it's to our benefit to do so (see above).
Something else to consider: If God were vain, one would think that he would want pictures and statues of him everywhere, yet he commanded that no one make images of him. Instead, he told the Israelites to keep copies of his commands everywhere (Dt 6:6-9), so that they would remember them and obey them and receive blessings as a result (Dt 6:18).
Why does God insist we worship only him?
Another objection is that it's unreasonable for God to demand that we worship him alone and no one else. Yet this too is for our benefit:
They pour out drink offerings to other gods to provoke me to anger. But am I the one they are provoking? declares the LORD. Are they not rather harming themselves, to their own shame? (Jer 7:18-19)
If Christianity is true, then the Christian God really is the only God, and praying to another god is praying to someone who doesn't exist. Not only is that counterproductive, it's harmful: the person who does so is ignoring the help and advice that God is willing to give him, and is instead seeking help from a nonexistent deity. At best, he will receive no benefit from the false god, and will have damaged his relationship with the true God (for praying to a false god is really saying to God, "I don't trust you to answer me or help me, so I'm going to seek help from another source," and also, "I don't believe you when you say you're the only God; your words aren't trustworthy"). At worst, praying to a false god allows him to be deceived - for instance, he may fool himself into thinking that the false god has given him permission to do something evil.
On the other hand, if we worship the true God, we receive all the benefits of a relationship with God, including his love, forgiveness and guidance. It follows that a trustworthy God who wants the best for us would instruct us to worship him and be upset if we didn't.