Thanks for your question Aaron. Christians worship God exclusively because the Bible makes it clear it is God who ‘created the heavens and the earth’. [Gen 1:1]. As a consequence, if God created all things, then there can be no other god. Any other god that people choose to worship is, in reality, something they have created or dreamt up for themselves.
You’ll notice that the various gods of Greeks, Romans and Egyptians usually related to aspects of creation like the sun, moon, sea etc. These are all things that God himself created. So God rightly calls on all people to worship him, and not the things that he created. Unfortunately, people choose to ignore God’s decree on this matter, which will bring about his judgement upon them [c.f. Romans 1:21-25].
In the Old Testament, this idea of exclusive worship of God was made explicit to Israel in the 1st & 2nd commandments [Exodus 20:3-6]. Here God entered into a covenant with his chosen people [Israel], in which they were obligated to worship him exclusively. We do the same when we enter into a marriage covenant, in that we promise to be exclusively faithful to our wife/husband and not enter into relationship with anyone else. As such, God demanded that his people not worship any other god or idol. The notion of exclusive worship of God by his people is famously declared in Deuteronomy 6:4 - "Hear, O Israel: The LORD our God, the LORD is one". Of course, the Old Testament revealed that Israel was constantly guilty of idolatry [Judges 2:11-15] and this eventually brought God’s judgement as they were exiled to Babylon and Assyria [2 Kings 17:13-23 & Jeremiah 25:1-9].
The New Testament continued to affirm that Christians worship one God. Jesus himself reaffirms Deut 6:4 that"the Lord is one" in Mark 12:29. The Apostle Paul also taught this [1 Corinthians 8:6; Galatians 3:20]. However the nature of the oneness of God shifted radically with the coming of Jesus. As the Son of God, Jesus taught his disciples that he and his Father were one [John 10:30, 17:11]. In addition, Jesus teaching about himself and the Holy Spirit [John 14:26; 15:26] demonstrated that God is one God, but three different persons - Father, Son and Holy Spirit [c.f. Matthew 28:19; 2 Corinthians 13:14].
This concept of “one God in three persons” or Trinity was a great difficulty for the early church. It took them 400 years to finally work it out!! The problem is that it makes no logical sense. Yet the bible is explicit about it - There is one God yet he is three persons. Not three gods united in one purpose, not three manifestations of the one God but one God in three persons. Ultimately our language cannot fully describe it and we are left with what the early church concluded in the creed:
We believe in one God
the Father, the Almighty, ...
We believe in one Lord, Jesus Christ,
the only Son of God,
eternally begotten of the Father,
God from God, Light from Light,
true God from true God,
begotten, not made,
of one Being with the Father; ...
We believe in the Holy Spirit, the Lord,
and the giver of life,
who proceeds from the Father and the Son,
who with the Father and the Son
is worshiped and glorified, ...
As a result, Christians today are still called to worship one God, through the Son - Jesus, as we are indwelt by the Holy Spirit.
I hope you find this helpful.