How can we refer to the first or the second decade the new century, a la 20s or 30s?
If you are referring to the first decade then you can use 'noughties'
There is no consensus on the best word to use as they do not flow as easily as 'twenties', 'thirties', etc.
The period from 1910-1919 has been referred to as the 'nineteen-tens' or 'teens', although there is no universally accepted term for the 2000-2009 decade.
'Twenty-tens' or 'Two thousand tens' is generally used at present however this is likely to change as this decade becomes referred to historically.
I’ve heard “the aughts” or “the two thousands” or “the twenty-o’s” for the decade 2000-2009.
Also, “the two thousand tens” or “the twenty-tens” for the decade 2010-2019.