To trust in God means to let go of all other gods and trust in him alone. To serve other gods alongside the one true God is to deny that he is sufficient.
In many places in South Africa, pastors have a constant battle with people wanting to practice their traditional African religion alongside Christianity. People are happy to come to church, but when there is a funeral or some other communal event, there is a great deal of pressure to get involved in rituals involving ancestors and other aspects of traditional African religion. This situation is very similar to what we read about in 1 Thessalonians, and we need to take the Thessalonian believers as our example: they turned from idols to serve the living and true God. Though it meant rejection and suffering, they would not compromise by holding on to false religion while attempting to worship the one true God.
9 For they themselves report concerning us the kind of reception we had among you, and how you turned to God from idols to serve the living and true God,