When did the new millennium begin?

Which “millennium”, are you referring to? It can be any 1,000 year period, there is no “official” millennium because the dictionary doesn’t establish so. In practice, we have two different millennias: 

The 3rd millennium = 2001–3000

The 2000s millennium = 2000–2999

It officially began on Monday, January 1, 2001.

Mon, 1 Jan 2001 until  Wed, 31 Dec 3000

It began in 2001 but society ignores it and says it began in 2000 because it’s like calling 2000 the ‘90s. I think this issue is very petty so if you want to go for the latter, I could care less.