Answer to old chestnut: millennium
The basic answer is at the International Date Line.
But the date line runs from pole to pole, and has a lot of little kinks in it. We can be more specific.
A lot of people celebrated Y2K parties in Sydney, Australia, wanting to be the first ones into the year 2000. But New Zealand is another two time zones over and saw Y2K begin sooner, 12 hours before GMT (actually, 13 hours ahead due to Daylight Savings Time in the Southern Hemisphere.
Beyond New Zealand are some Pacific island nations that are close to the date line. Chatham Island is in an odd time zone 45 minutes ahead of New Zealand, and Tonga is an hour ahead of New Zealand. For many years, Tonga was the first time zone to see each new day, 13 hours before GMT.
However, more recently the president of the Kiribati Islands (an island nation spread across three time zones, and which had until that time been split by the Date Line) made a proclamation declaring that all of his nation would be on the western side of the Date Line, effectively decreeing a huge wild deviation into the Date Line itself!
Since each nation has the right to define its own time zones, this is fully valid, and so now the Line Islands of Kiribati will be the first time zone to enter each new day, 14 hours ahead of GMT (15 hours during daylight savings time).