Many refer correctly to Jesus as a master teacher. Not only was Jesus able to communicate effectively to those who had ears to hear, but he always recognized those "teachable moments" when they presented themselves. Some examples are when he taught about living water to the woman at the well (John 4), or about the bread of life when he fed the crowds (John 6).
We see the same sort of thing in the ministry of Paul. When he arrived in Athens, he used an altar to an unknown God as an opportunity to teach about the one true God (Acts 17:16-34). In fact, Paul even quotes from their own poets, "For we also are his children (Acts 17:28)." Originally, this quotation referred to one of the pagan gods. But Paul uses it as an opportunity to teach about the true God.
In fact, Paul said,
"…I have become all things to all men, so that I may by all means save some. I do all things for the sake of the gospel, so that I may become a fellow partaker of it (1 Cor 9:22-23)."
Paul sought to understand pagans and Jews alike in order to teach them the Gospel more effectively. He understood that he was not of the world. This didn't mean that he was called out of the world, but that he was called to go into the world with the Gospel.
The same holds true for us. Our citizenship is in Heaven (Phil 3:20), which is why we are called to be "ambassadors for Christ (2 Cor 5:20)." This takes discernment to balance both not being "conformed to this world (Rom 12:2), and being ambassadors for Christ.
How does this apply when it comes to religious holidays that have no basis in scripture? We have days on the Christian calendar such as Valentine's Day, Easter, Thanksgiving, and Christmas. Is it better to ignore these things or to participate in them? Instead of merely ignoring them or participating in them uncritically, it is better to identify teaching or ministry opportunities in them.
There is an example of this in the ministry of Jesus (John 10:22-42). There was a festival called the "Feast of Dedication" which commemorated the purification and rededication of the temple during the intertestamental period. The Temple was overtaken by the pagan ruler, Antiocus Epiphanes. He defiled it with pagan worship, the most offensive being the offering of a pig on the altar. Judas Maccabees revolted, retook the temple, consecrated and rededicated it in 164 A.D. The Feast of Dedication, which is now called "Hanukkah" commemorates this. You will not find this anywhere in your Bibles. As I understand it, this has some cultural baggage with roots in paganism just as Christmas does.
During this feast, Jesus neither ignores it nor does he boycott and speak out against it as a tradition of men. Instead, he uses it as an opportunity to teach an important truth about himself that he is the Son of God.
We shouldn't just uncritically "go with the flow" of the world. Instead, we ought always to be looking for opportunities to use what is going on around us as a ministry opportunity as Jesus did.