Yes, serving God is important; it is our response to Him. Recognize, though, that God did not create us for the purpose of serving Him as if He needs something (Acts 17:24–25). God created us in His image and for relationship with Him and one another (Genesis 1:26–28). We display the character and work of God to one another and the rest of creation (Matthew 5:14–16; Ephesians 3:9–11). God also invites us to participate in His work in meaningful ways (Genesis 1:28; 9:1; John 15:15–17). For those who are in Christ, we are in intimate relationship with the Trinity and one another (John 17:20–26; Ephesians 4), and we will one day dwell with God (Revelation 21:1–4). In healthy relationships we serve one another. Jesus gave us the example of serving by serving His disciples (cf. Mark 10:43–45; John 13:12–17), and when we serve others, we are serving the Lord (Matthew 25:35–40). He demonstrated that we are to seek the welfare of others first (Philippians 2:1–11). The Spirit of God has given His people gifts with which to serve and build up the body of Christ (1 Corinthians 12; Romans 12:3–8).
Our response to the Lord God creating us should be to serve Him. And serving Him includes serving others. In what is commonly called the Upper Room Discourse, Jesus used His example of washing His disciples' feet to show that we should engage in service to others. Jesus was telling His disciples that service is important to God. Jesus also stated elsewhere that the order of importance in God's kingdom is not the same as the order of importance in man’s world. Service is important to God, so it should be important to us.
In providing for His church, God gave us gifts—not for our own uplifting but for building up the church. God has not only created us because He wanted to, but He has equipped us to advance according to His standards. Improving our own standing is not what He desires, but we should work for the betterment of others.
So this raises the question of how serving God translates into serving other people. If we are to serve God, we must know and follow His standards. God's standard of greatness is not the same as ours. His standard, that of love, puts others first (1 Corinthians 13), so we should put others first to follow His standards. In serving God, we will serve other people, and in serving others, we serve God.