The New Year, or novus annus, was celebrated by decking the house with olive, laurel, or myrtle branches for the Kalends of January. (Prior to 153 BC, the new year began on the first of March.) Sacrifices were made to Janus; gifts and visits were exchanged, and masquerading and feasting were the general activities. Congratulatory presents were made to the magistrates who entered upon office on this day, and the emperors exacted tribute of a pound of gold from their subjects. This quasi-present was called strena, a term that was extended to all New Year's gifts in Rome.
Traditionally it derived from a custom initiated by the legendary King Tatius, to whom branches of vervain gathered in the sacred Grove of Strenua, the goddess of health and physical vigor, were presented as a good omen on the first day of the year 747 BC. (The imperial strenae later became so excessive that Claudius found it necessary to limit the amount by formal decree.) Other gifts included dates, figs, or honey so that, according to Ovid, the year might go in sweetness as it had begun. Wax tapers and oil lamps, tied up with bits of holly, were also considered appropriate gifts for friends to light the way in the coming year.
A curious New Year's practice in the village of Caerleon-on-Usk might be the relic of a custom introduced by a Roman legion garrisoned there centuries earlier.
Some of the preceding information comes from Handbook to Life in Ancient Rome, written by Lesley Adkins and Roy A. Adkins and published in1994 by The Oxford University Press (New York).