To me, it was one of the first indications of how the media can massively hype something up, and cause big panic, or big sensation, and I have been sceptical of anything "big" in the media ever since, and started to notice how they deliberately cause a stir. Indeed, the "Millennium" itself was a load of hype: Millennium this, Millennium that, and that utterly wasteful Millennium Dome. My parents also explained to me about spin doctors, and for an example, they said: let us suppose that there was some massive terrorist plot due to happen on 1st January 2000, while the police were busy, which was successfully stopped, but the government didn't want to the public to know about: they would have made the headlines said "Queen drops toffee paper in Dome" or something.
I expect there was indeed lots of work behind the scenes to put it right, which was barely reported on, as things going right usually doesn't make the news. But at the time, Tony Blair was talking about how he was going to organise an army to fix it. My parents encouraged me to write to his office, to say I wanted to be part of it. The ambiguous reply I received implied that there was no such government scheme at all, or maybe I was too late for it.
From what I remember reading at the time, some of the problems started a few years before, when credit cards started having expiry dates of 00. I've also heard that there might be other, less foreseeable problems down the line: many systems store the date as "number of seconds since a certain date", such as 1st January 1901, and this number might become too big for the computer. There's also a calendar curiosity that although 2000 was a leap year, years ending in "00" are only leap years if they are divisible by 400, so 1800, 1900 and 2100 are not leap years. Some computer systems got round this by only allowing dates between 1901 and 2099. Not that many of us will be alive in 2100, but it could matter to somebody.
It also makes me smile that calendar problems are centuries old, such as when the Julian calendar was considered not accurate enough, and people rioted about their lives being shortened by eleven days when the change was made.
In the film "Entrapment", the Millennium Bug is exploited by thieves who steal a lot of money from International Clearance Bank when it is shut down for thirty seconds at midnight, because of the Millennium Bug.