In the earliest days of the Church (the first and second centuries after Christ) there were no such celebrations as Christmas and New Year’s Day.
The New Catholic Encyclopedia states:
“According to the hypothesis … accepted by most scholars today, the birth of Christ was assigned the date of the winter solstice (December 25 in the Julian [Roman] Calendar, January 6 in the Egyptian), because on this day, as the sun began to return to northern skies, the pagan devotees of Mithra celebrated dies natalis Solis Invicti (birthday of the invincible sun)” (1967, Vol. 3, p. 656).
Those customs carried over in the observance of Christmas (with its many traditions and practices steeped in paganism), and the “birth” of the “new year” of the sun. This is why the Roman calendar in use today designates Jan. 1 as the beginning of the new year as opposed to God’s designation of the spring month of Abib or Nisan on the Hebrew calendar as the beginning of the year (Exodus 12:1-2 Exodus 12:1-2  And the LORD spoke to Moses and Aaron in the land of Egypt saying,
 This month shall be to you the beginning of months: it shall be the first month of the year to you.
American King James Version×).
Much of the symbolism associated with New Year’s celebrations today has very definite pagan origins. Kissing at the moment of transition to the next year is rooted in pagan sexual practice and superstition. And evergreen wreaths associated with Christmas were originally part of the pagan Calend celebration of Jan. 1.
Because we live in this secular society and most of the world operates according to the commonly used Roman calendar, it’s common to think of a new calendar year as beginning Jan. 1. Yet involvement in the celebration of that is not appropriate.
That said, one should be careful not to become too judgmental if, for example, someone is viewing a football game or parade at that time of year—as opposed to actually celebrating a New Year festival.
If you’d like to learn more, many resources are available that show the pagan origins of New Year’s Day celebrations. You can find related material in reference encyclopedias and online research resources as well.