No, today is not the new year by any reckoning I know. Most people call January 1 the start of the new year, but really it depends on the calendar you use. The vernal equinox always seemed like a more meaningful choice to me, but though the equinox is a global phenomenon, spring is not, so maybe an arbitrary day is as good as any.
The Chinese new year, celebrated by perhaps 1.3 billion people, is coming up on January 26, and it's the last in a run of major new year holidays. I often hear people call it "the lunar new year", which always bugs me because the Chinese calendar is just one of many popular lunar calendars, and because I'm a pedantic nitpicker. (And really, if we're going to nitpick, the Chinese calendar is not strictly lunar, because it tracks the tropical solar year.)
Around 1.5 billion people follow the Islamic calendar, the only truly lunar calendar I've ever heard of (probably because lunar calendars are not useful for any practical purpose), which celebrated its new year on December 28. September 29 marked the Jewish New Year, celebrated by a vocal minority of 15 million.
(Population estimates gleaned from the always-reliable internet.)