I can't specifically remember any new years eves as a kid, though I know I stayed up with the family and watched (in black-and-white) the ball drop in Times Square.
I do remember where I was on New Years Eve, 1999 - I was freezing down in San Pedro, by the Korean Bell, a gift from South Korea to the City of Los Angeles., that sits on the coast. I was helping out as a cameraman on a live broadcast to South Korea. It was freezing, I had a cold - it was miserable! But we had a little excitement when our power generators went out ten minutes before midnight, not only leaving all of the crew in the dark, but all of the guests as well (a crowd of dignitaries were to ring in the new year en masse using the large bell. It turns out the production crew was providing all of the light for the event as well. Just in time, all the power was restored and the show went on as scheduled.
I've been to numerous New Years Eve parties over the years, but the truth is that most are the same sort of thing - I'm not one for drunken all-out parties, and many of the others I've gone to are otherwise uneventful - hardly worth risking the drunk drivers out on the roads!
As for Times Square, I can't help but marvel at those people who brave a cold December day in New York City to stand around Times Square for most of the day waiting for the big moment - which is over in perhaps ten or fifteen minutes (though I have to admit there is some excitement in just "being there."
The other thought that almost always occurs to me is the transitory nature of new years eve - the fact that it happens in waves throughout the day, from one side of the world to the other - while you're celebrating, people in Australia have already moved on with their lives. Being in California, we're just about at the end of the process - like so many other live programs, we see Times Square on a three-hour delay. Since we don't have any great televised New Year celebration, we're stuck with leftovers...