The Church's New Year begins in Advent.
We do not have a liturgical celebration for the astronomical New Year.
January 1 is the Solemnity of Mary, Mother of God.
We have very apt readings for today's Mass. The end of the year reminds the Church of the end of the world: "Children, it is the last hour; and just as you heard that the antichrist was coming, so now many antichrists have appeared" (1 Jn 2:18). The gospel goes back to the beginning, with John's new Genesis account: "In the beginning was the Word, and the Word was with God, and the Word was God" (Jn 1:1).
May God bless our memories of our last pilgrimage around the sun and grant all of our best wishes for our next.
New Year's Resolutions
Just as we have to be on guard against the secularization of Christmas, so we need to guard against the secularization of New Year's Day. God's New Year is different from the astronomical New Year.
That said, this is the one time in the year that our culture invites us to examine our consciences, review the year past, and set ourselves to cultivate better behavior in the New Year. Those good impulses are something that God's Spirit can use to touch our hearts.
We act our way into new feelings rather than feel our way into new actions. We are creatures of habit. Our New Year's resolutions are all about changing habits.
Habits are practical skills (virtues) that make doing good easy and doing evil difficult.
We develop a habit by practice; they don't come overnight and, at least so far in my experience, aren't normally delivered by a Habit Fairy overnight.
We learn by doing. We learned how to walk by walking poorly at first, how to talk by talking poorly at first, and how to believe by believing poorly at first. "I believe; help Thou my unbelief." We have to take small steps at first; leaps and bounds come later, if we have laid a good foundation.