Ancient Babylonians are credited with the earliest celebrations of "new year" with a religious festival called Akitu (meaning barley) in late March.
It made sense that "New Years" was celebrated when Spring brought in good weather and initiated the cycle of planting/harvesting new crops. Some countries celebrated specific annual events (like the Egyptian Nile flooding) and/or astrological events (such as the Chinese honoring the second new moon after winter solstice).
Ancient Romans continued to celebrate spring or March (The Festival of Calends) as the new year until they added to their Julian Calendar the month of "January", named after the god Janus. Janus, the god of doors or beginnings, had two heads - to look backwards (on the past) and to look forward (on the future). As a side note, the original Roman calendar only had 10 months and it was frequently altered by new rulers.
Even before the Gregorian calendar was adopted, which replaced the older Julian calendar, most people were celebrating New Years Day on January 1st. A few countries/areas continued to celebrate "new year" in Spring (with the Feast of Annunciation) until the Gregorian calendar was officially adopted in 1752. Some countries celebrated new year after summer. One such celebration is called Samhain or Summer's End.
People celebrated New Year (regardless of the calendar used or the actual date) with sweets, gift exchanges, and feasts. In Rome, a festival was held that lasted three days and even allowed slaves to participate. Typically, slaves had their own holiday/celebration days separate from the citizens, but this was one of the times of the year where they celebrated together regardless of citizenship. It was a time to renew friendships and put aside past transgressions. Gifts were given to the emperor (more lavish gifts than normally bestowed) in return for favors. This practice also became popular in other countries, such as Persia.
In 487 AD, New Year's Day was declared the Feast of the Circumcision, the eighth day after Christ was born, a solemn Christian holiday that disallowed parties or heathenist celebrations. Over time, however, festivities returned, especially in the middle ages.
The act of making New Year resolutions is believed to have started when ancient Babylonias promised to return borrowed tools to their rightful owners. English people cleaned their chimneys on New Years day, which eventually morphed into the idea of "cleaning the slate", the act of resolving to start afresh in the new year.
Wikipedia - New Years Day
History.com - New Years
The Traditions of New Year