When did people start celebrating the new years (within the last 2016 years)? Did they count down every new years eve starting in the year 1 or was it much later?

There have been (and still are) many different celebrations of the new year.

I believe the oldest that we know of currently was Akitu (celebrated by the Sumerians). This was a set of rituals that took place over many days in what is now March for us.

The earliest Romans also celebrated the new year in March, but 153 B.C.E. was the first time January 1st was designated as the beginning of the new year.

It's a relatively recent change that has most countries accepting January 1st as the new year. For example, in England it wasn't until around 1750.