I covered the Old Testament understanding of not having other gods and not making any graven images, in Molten Gods last week. After the Captivity in Babylon those were no longer an issue, though they were still used for divination. Divination continued to be a problem up until Jesus's time, and continuing on into our days as well.
From the Catholic Catechism: All forms of divination are to be
rejected: recourse to Satan or demons, conjuring up the dead or other
practices falsely supposed to "unveil" the future. Consulting
horoscopes, astrology, palm reading, interpretation of omens and lots,
the phenomena of clairvoyance, and recourse to mediums all conceal a
desire for power over time, history, and, in the last analysis, other
human beings, as well as a wish to conciliate hidden powers. (Emphasis mine. Oh well, there goes my preferred superpower of time travel. Though I suppose any superpower might have the same effect.)
There are a few things that get described as other gods in the NT. I could make the argument that the Book of Jude is largely a warning that worshiping other gods leads to disobeying all the other Declarations.*
Jesus identifies money, or Mammon, as a rival god. Matthew 6:24-34
v. 24 No one
can serve two masters. Either you will hate the one and love the other,
or you will be devoted to the one and despise the other. You cannot
serve both God and money. (Also Luke 16:13)
We might regard wealth as something of a cross between a physical and an abstract God, but Jesus names him and makes it sound personal, as if he were talking about Zeus or Apollo. Perhaps he was only using it as a common metaphor, but I wouldn't want to count on that. It might be someone He knew personally.
Paul is talking about the more familiar types of gods that we see in books for students about "Myths of Many Lands" or the like when he speaks in the agora. Acts: 17:24-31. Key verse is 29, Therefore since we are
God’s offspring, we should not think that the divine being is like gold
or silver or stone—an image made by human design and skill. But that does hearken back to the household gods of the OT as well.
The Epistles in general go on to describe gods more in the way we are used to: sets of behaviors, especially greedy or sexual, that take over our lives and become "gods" to us, even if they aren't named Venus or Freya. (Though Revelation makes a strong tie-in with sacrificing to other named gods and sexual immorality Revelation 2:14 Revelation 2:20)
Put to death, therefore, whatever belongs to your earthly nature: sexual immorality, impurity, lust, evil desires and greed, which is idolatry.
Galatians 5:19-21 In part
The acts of the flesh are obvious: sexual immorality, impurity and debauchery; 20 idolatry and witchcraft
Their destiny is destruction, their god is their stomach, and their glory is in their shame. Their mind is set on earthly things.
Ephesians 5:3-6 v.5
For of this you can be sure: No immoral, impure or greedy person—such a person is an idolater—has any inheritance in the kingdom of Christ and of God.
As I mentioned before, the commandment against adultery may not refer only to family harmony, keeping promises, and being loyal, but to refraining from worshiping other gods. The whole town may be going out to do that for festival, and they may think their fertility cause for their flocks and fields is jeopardised by your abstaining, but you are not to join them.
*Eventually the whole list comes in.
James 4:3-5 Therefore, anyone who chooses to be a friend of the world becomes an enemy of God
I think that covers the territory. In contrast, God has chosen to speak, not through idols, or temples, or even magical ceremonies, but in words, actions in history, and actions of his people.
So here's the exercise. What gods are behind the other commandments? Stealing is connected to wealth, certainly - though security, laziness, or revenge could also be motives.
What god is worshiped when we bear false witness, or don't keep the Sabbath? Scratch them down. It's a fun, and I hope revealing exercise.